A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)*(n+k+1)*binomial(n,k)*binomial(n+k,k) for n >= k >= 0.

1, -2, 6, 3, -24, 30, -4, 60, -180, 140, 5, -120, 630, -1120, 630, -6, 210, -1680, 5040, -6300, 2772, 7, -336, 3780, -16800, 34650, -33264, 12012, -8, 504, -7560, 46200, -138600, 216216, -168168, 51480, 9, -720, 13860, -110880, 450450, -1009008, 1261260, -823680, 218790

Offset

0,2

Comments

Tables I, III, IV on pages 92 and 93 of Ser have integer entries and are A331430, A331431 (the present sequence), and A331432.

Given the system of equations 1 = Sum_{j=0..n} H(i, j) * x(j) for i = 2..n+2 where H(i,j) = 1/(i+j-1) for 1 <= i,j <= n is the n X n Hilbert matrix, then the solutions are x(j) = T(n, j). - Michael Somos, Mar 20 2020 [Corrected by Petros Hadjicostas, Jul 09 2020]

References

J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93. See Table III.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

A. Buhl, Book review: J. Ser - Les calculs formels des séries de factorielles, L'Enseignement Mathématique, 32 (1933), p. 275.

L. A. MacColl, Review: J. Ser, Les calculs formels des séries de factorielles, Bull. Amer. Math. Soc., 41(3) (1935), p. 174.

L. M. Milne-Thomson, Review of Les calculs formels des séries de factorielles. By J. Ser. Pp. vii, 98. 20 fr. 1933. (Gauthier-Villars), The Mathematical Gazette, Vol. 18, No. 228 (May, 1934), pp. 136-137.

J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].

J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages.)

Formula

T(n, 0) = (-1)^n*A000027(n+1).

T(n, 1) = A331433(n-1) = (-1)^(n+1)*A007531(n+2).

T(n, 2) = A331434(n-2) = (-1)^n*A054559(n+3).

T(n, n-2) = A002738(n-2).

T(n, n-1) = (-1)*A002736(n).

T(n, n) = A002457(n).

T(2*n, n) = (-1)^n*(3*n+1)!/(n!)^3 = (-1)^n*A331322(n).

Sum_{k=0..n} T(n, k) = A000290(n+1) (row sums).

Sum_{k=0..n}((-1)^k*T(n, k) = (-1)^n*A108666(n+1) (alternating row sums).

Sum_{k=0..n} T(n-k, k) = (-1)^n*A109188(n+1) (diagonal sums).

2^n*Sum_{k=0..n} T(n, k)/2^k = (-1)^floor(n/2)*A100071(n+1) (positive half sums).

(-2)^n*Sum_{k=0..n} T(n, k)/(-2)^k = A331323(n) (negative half sums).

T(n, k) = ((2*k+1)!/(k!)^2)*[x^(n-k)] (1+x)^(-2*(k+1)). - Georg Fischer and Peter Luschny, Jan 18 2020

T(n,k) = (-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!), for n >= k >= 0. - N. J. A. Sloane, Jan 18 2020

From Petros Hadjicostas, Jul 09 2020: (Start)

Michael Somos's formulas above can be restated as

Sum_{k=0..n} T(n,k)/(i+k) = 1 for i = 1..n+1.

These are special cases of the following formula that is alluded to (in some way) in Ser's book:

1 - Sum_{k=0..n} T(n,k)/(x + k) = (x-1)*...*(x-(n + 1))/(x*(x+1)*...*(x+n)).

Because T(n,k) = (-1)^(n+1)*(n + k + 1)*A331430(n,k) and Sum_{k=0..n} A331430(n,k) = (-1)^(n+1), one may derive this formula from Ser's second formula stated in A331430. (End)

T(2*n+1, n) = (-2)*(-27)^n*Pochhammer(4/3, n)*Pochhammer(5/3, n)/(n!*(n+1)!). - G. C. Greubel, Mar 22 2022

Example

Triangle begins:
   1;
  -2,    6;
   3,  -24,    30;
  -4,   60,  -180,     140;
   5, -120,   630,   -1120,     630;
  -6,  210, -1680,    5040,   -6300,     2772;
   7, -336,  3780,  -16800,   34650,   -33264,   12012;
  -8,  504, -7560,   46200, -138600,   216216, -168168,   51480;
   9, -720, 13860, -110880,  450450, -1009008, 1261260, -823680, 218790;
  ...

Maple

gf := k -> (1+x)^(-2*(k+1)): ser := k -> series(gf(k), x, 32):
T := (n, k) -> ((2*k+1)!/(k!)^2)*coeff(ser(k), x, n-k):
seq(seq(T(n, k), k=0..n), n=0..7); # Peter Luschny, Jan 18 2020
S:=(n, k)->(-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!);
rho:=n->[seq(S(n, k), k=0..n)];
for n from 0 to 14 do lprint(rho(n)); od: # N. J. A. Sloane, Jan 18 2020

Mathematica

Table[(-1)^(n+k)*(n+k+1)*Binomial[2*k, k]*Binomial[n+k, n-k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 22 2022 *)

Prog

(Magma) [(-1)^(n+k)*(k+1)*(2*k+1)*Binomial(n+k+1, n-k)*Catalan(k): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 22 2022
(Sage) flatten([[(-1)^(n+k)*(2*k+1)*binomial(2*k, k)*binomial(n+k+1, n-k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Mar 22 2022

Crossrefs

Columns 1 is A331433 or equally A007531, column 2 is A331434 or equally A054559; the last three diagonals are A002738, A002736, A002457.

Cf. A000290 (row sums), A002457,, A100071, A108666 (alternating row sums), A109188 (diagonal sums), A331322, A331323, A331430, A331432.

Sequence in context: A364318 A008306 A231171 * A248120 A144362 A303761

Adjacent sequences: A331428 A331429 A331430 * A331432 A331433 A331434

Keyword

sign,tabl

Author

N. J. A. Sloane, Jan 17 2020

Extensions

Several typos in the data corrected by Georg Fischer and Peter Luschny, Jan 18 2020

Definition changed by N. J. A. Sloane, Jan 18 2020

Status

approved