A266521 E.g.f.: Log( Sum_{n>=0} (n + y)^(2*n) * x^n/n! ) = Sum_{n>=1} Sum_{k=0..n+1} T(n,k) * x^n*y^k/n!, as a triangle of coefficients T(n,k) read by rows.

1, 2, 1, 15, 28, 18, 4, 683, 1278, 933, 316, 42, 62038, 117440, 92680, 38240, 8272, 752, 9342629, 17880090, 14855385, 6881640, 1880340, 288048, 19360, 2100483216, 4054752672, 3490688496, 1743156480, 547098240, 108228192, 12523584, 654912, 658746323647, 1279910119670, 1130161189549, 594323331364, 204256939502, 47125635760, 7147508032, 652959872, 27546736, 274730459045232, 536368375356928, 482514140459520, 263340552849920, 96404466197760, 24628940050176, 4404380994048, 533057051648, 39701769216, 1388207872

Offset

1,2

Comments

Row sums form A266520, coefficients in Log( Sum_{n>=0} (n+1)^(2*n) * x^n/n! ).

Column 0 forms A266519, coefficients in log( Sum_{n>=0} n^(2*n) * x^n/n! ).

Rightmost border is A266526.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1125 as a flattened triangle read by rows 1..45.

Example

E.g.f.: A(x,y) = x * (1 + 2*y + y^2) +
x^2/2! * (15 + 28*y + 18*y^2 + 4*y^3) +
x^3/3! * (683 + 1278*y + 933*y^2 + 316*y^3 + 42*y^4) +
x^4/4! * (62038 + 117440*y + 92680*y^2 + 38240*y^3 + 8272*y^4 + 752*y^5) +
x^5/5! * (9342629 + 17880090*y + 14855385*y^2 + 6881640*y^3 + 1880340*y^4 + 288048*y^5 + 19360*y^6) +
x^6/6! * (2100483216 + 4054752672*y + 3490688496*y^2 + 1743156480*y^3 + 547098240*y^4 + 108228192*y^5 + 12523584*y^6 + 654912*y^7) +...
where
exp(A(x,y)) = 1 + (1 + y)*x + (2 + y)^4*x^2/2! + (3 + y)^6*x^3/3! + (4 + y)^8*x^4/4! + (5 + y)^10*x^5/5! + (6 + y)^12*x^6/6! +...
This triangle begins:
1, 2, 1;
15, 28, 18, 4;
683, 1278, 933, 316, 42;
62038, 117440, 92680, 38240, 8272, 752;
9342629, 17880090, 14855385, 6881640, 1880340, 288048, 19360;
2100483216, 4054752672, 3490688496, 1743156480, 547098240, 108228192, 12523584, 654912;
658746323647, 1279910119670, 1130161189549, 594323331364, 204256939502, 47125635760, 7147508032, 652959872, 27546736;
274730459045232, 536368375356928, 482514140459520, 263340552849920, 96404466197760, 24628940050176, 4404380994048, 533057051648, 39701769216, 1388207872;
147034646085347145, 288100398039817266, 262835789583073329, 147457696629622032, 56514667400140392, 15510808994500512, 3097157140510272, 445604738641920, 44324678623680, 2758053332736, 81621893376; ...

Prog

(PARI) {T(n, k) = n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^(2*m) *x^m/m! ) +x*O(x^n) ), n, x), k, y)}
for(n=1, 10, for(k=0, n+1, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A266519, A266520, A266526.

Sequence in context: A219899 A326600 A272304 * A039652 A012894 A013076

Adjacent sequences: A266518 A266519 A266520 * A266522 A266523 A266524

Keyword

nonn,tabf

Author

Paul D. Hanna, Jan 01 2016

Status

approved