A359670 Triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) satisfying y = Sum_{n=-oo..+oo} (-1)^n * x^n * (y*A(x,y) + x^(n-1))^(n+1).

1, 2, 1, 4, 6, 1, 8, 21, 12, 1, 14, 62, 68, 20, 1, 24, 162, 284, 170, 30, 1, 40, 384, 998, 970, 360, 42, 1, 64, 855, 3092, 4410, 2720, 679, 56, 1, 100, 1806, 8724, 17172, 15627, 6608, 1176, 72, 1, 154, 3648, 22904, 59545, 74682, 47089, 14392, 1908, 90, 1, 232, 7110, 56679, 188700, 311530, 271698, 125160, 28764, 2940, 110, 1

Offset

0,2

Comments

Related identity: 0 = Sum_{-oo..+oo} (-1)^n * x^n * (y + x^n)^n, which holds formally for all y.

T(n,0) = A015128(n), the number of overpartitions of n, for n >= 0.

T(n+1,1) = A022571(n), the coefficient of x^n in Product_{m>=1} (1 + x^m)^6, for n >= 0.

A359711(n) = Sum_{k=0..n} T(n,k) for n >= 0 (row sums).

A359712(n) = Sum_{k=0..n} T(n,k)*2^k for n >= 0.

A359713(n) = Sum_{k=0..n} T(n,k)*3^k for n >= 0.

A363104(n) = Sum_{k=0..n} T(n,k)*4^k for n >= 0.

A363105(n) = Sum_{k=0..n} T(n,k)*5^k for n >= 0.

A359714(n) = T(2*n,n) for n >= 0 (central terms).

A359715(n) = T(n+2,2) for n >= 0.

A359718(n) = T(n+3,3) for n >= 0.

A363142(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) for n >= 0. - Paul D. Hanna, May 18 2023

From Paul D. Hanna, May 20 2023: (Start)

A363182(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) * 2^(n-2*k) for n >= 0.

A363183(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) * 3^(n-2*k) for n >= 0.

A363184(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) * 4^(n-2*k) for n >= 0.

A363185(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) * 5^(n-2*k) for n >= 0. (End)

Links

Paul D. Hanna, Table of n, a(n) for n = 0..2555

Formula

G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^n*y^k may be described as follows.

(1) y = Sum_{n=-oo..+oo} (-1)^n * x^n * (y*A(x,y) + x^(n-1))^(n+1).

(2) x*y = Sum_{n=-oo..+oo} (-1)^n * (x*y*A(x,y) + x^n)^(n+1).

(3) x*y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*A(x,y)*x^(n+1))^(n-1).

(4) A(x,y) = 1/[Sum_{n=-oo..+oo} (-1)^n * (x*y*A(x,y) + x^n)^n ].

(5) A(x,y) = 1/[Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + y*A(x,y)*x^(n+1))^n ].

From Paul D. Hanna, May 18 2023: (Start)

(6) y = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (y*A(x,y) + x^n)^n.

(7) A(x,y) = 1/[Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (y*A(x,y) + x^n)^n ].

(8) x*y = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + y*A(x,y)*x^(n+1))^(n+1).

(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (y*A(x,y) + x^n)^(n+1).

(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*A(x,y)*x^n)^n.

(11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*A(x,y)*x^(n+1))^n. (End)

Example

G.f.: A(x,y) = 1 + x*(2 + y) + x^2*(4 + 6*y + y^2) + x^3*(8 + 21*y + 12*y^2 + y^3) + x^4*(14 + 62*y + 68*y^2 + 20*y^3 + y^4) + x^5*(24 + 162*y + 284*y^2 + 170*y^3 + 30*y^4 + y^5) + x^6*(40 + 384*y + 998*y^2 + 970*y^3 + 360*y^4 + 42*y^5 + y^6) + x^7*(64 + 855*y + 3092*y^2 + 4410*y^3 + 2720*y^4 + 679*y^5 + 56*y^6 + y^7) + x^8*(100 + 1806*y + 8724*y^2 + 17172*y^3 + 15627*y^4 + 6608*y^5 + 1176*y^6 + 72*y^7 + y^8) + x^9*(154 + 3648*y + 22904*y^2 + 59545*y^3 + 74682*y^4 + 47089*y^5 + 14392*y^6 + 1908*y^7 + 90*y^8 + y^9) + x^10*(232 + 7110*y + 56679*y^2 + 188700*y^3 + 311530*y^4 + 271698*y^5 + 125160*y^6 + 28764*y^7 + 2940*y^8 + 110*y^9 + y^10) + ...
This triangle of coefficients T(n,k) of x^n*y^k in A(x,y), for n >= 0, k = 0..n, begins
[1];
[2, 1];
[4, 6, 1];
[8, 21, 12, 1];
[14, 62, 68, 20, 1];
[24, 162, 284, 170, 30, 1];
[40, 384, 998, 970, 360, 42, 1];
[64, 855, 3092, 4410, 2720, 679, 56, 1];
[100, 1806, 8724, 17172, 15627, 6608, 1176, 72, 1];
[154, 3648, 22904, 59545, 74682, 47089, 14392, 1908, 90, 1];
[232, 7110, 56679, 188700, 311530, 271698, 125160, 28764, 2940, 110, 1];
[344, 13434, 133516, 556085, 1169100, 1342684, 860664, 300888, 53640, 4345, 132, 1];
[504, 24702, 301664, 1542640, 4029237, 5884160, 4980320, 2438712, 666240, 94490, 6204, 156, 1];
[728, 44361, 657368, 4065868, 12940766, 23411339, 25215416, 16367874, 6302148, 1377464, 158708, 8606, 182, 1];
[1040, 78006, 1387854, 10253720, 39153924, 85994062, 114672768, 94919382, 48660900, 15071628, 2687454, 256022, 11648, 210, 1]; ...
RELATED SERIES.
Given g.f. F(x) of A361770, where
F(x) = 1 + 3*x + 14*x^2 + 80*x^3 + 510*x^4 + 3498*x^5 + 25145*x^6 + 186972*x^7 + 1426159*x^8 + 11096944*x^9 + 87736474*x^10 + ... + A361770(n)*x^n + ...
then
(1) F(x) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k) * F(x)^k,
(2) F(x) = Sum_{n=-oo..+oo} (-1)^n * x^n * (F(x)^2 + x^(n-1))^(n+1).
Given g.f. G(x) of A363135, where
G(x) = 1 + 3*x + 17*x^2 + 133*x^3 + 1201*x^4 + 11796*x^5 + 122192*x^6 + 1314266*x^7 + 14536760*x^8 + 164299909*x^9 + ... + A363135(n)*x^n + ...
then
(1) G(x) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k) * G(x)^(2*k),
(2) G(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (F(x)^3 + x^(n-1))^(n+1).

Prog

(PARI) {T(n, k) = my(A=1); for(i=1, n,
A = 1/sum(m=-#A, #A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( polcoeff( A, n, x), k, y)}
for(n=0, 15, for(k=0, n, print1( T(n, k), ", ")); print(""))
(PARI) {T(n, k) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-y + sum(n=-#A, #A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y), #A-1, x) ); polcoeff( A[n+1], k, y)}
for(n=0, 15, for(k=0, n, print1( T(n, k), ", ")); print(""))

Crossrefs

Cf. A359711 (row sums), A359712 (y=2), A359713 (y=3), A363104(y=4), A363105 (y=5).

Cf. A359714 (central terms), A359715 (column 2), A359718 (column 3).

Cf. A363142, A363182, A363183, A363184, A363185.

Cf. A361770, A363135, A363136, A363137.

Cf. A359720, A293600.

Sequence in context: A181854 A109822 A274292 * A114192 A114656 A294440

Adjacent sequences: A359667 A359668 A359669 * A359671 A359672 A359673

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 17 2023

Status

approved