A331323 -id:A331323 - cp's OEIS frontend

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

N. J. A. Sloane, Jan 17 2020

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]

			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;
  ...

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

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.)

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.

[(-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

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

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 *)

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

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

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

A331511 Square array T(n,k), n >= 0, k >= 0, read by descending antidiagonals, where column k is the expansion of (1 - (k-3)x)/(1 - 2(k-1)x + ((k-3)x)^2)^(3/2).

Original entry on oeis.org

1, 1, 0, 1, 2, -15, 1, 4, -6, 32, 1, 6, 9, -12, 105, 1, 8, 30, 16, 30, -576, 1, 10, 57, 140, 25, 60, 105, 1, 12, 90, 384, 630, 36, -140, 5760, 1, 14, 129, 772, 2505, 2772, 49, -280, -13167, 1, 16, 174, 1328, 6430, 16008, 12012, 64, 630, -30400

Offset: 0

Seiichi Manyama, Jan 18 2020

			Square array begins:
      1,   1,  1,    1,     1,     1, ...
      0,   2,  4,    6,     8,    10, ...
    -15,  -6,  9,   30,    57,    90, ...
     32, -12, 16,  140,   384,   772, ...
    105,  30, 25,  630,  2505,  6430, ...
   -576,  60, 36, 2772, 16008, 52524, ...
.
From _Peter Luschny_, Jan 20 2020: (Start)
Read by ascending antidiagonals gives:
[0]      1
[1]      0,    1
[2]    -15,    2,  1
[3]     32,   -6,  4,     1
[4]    105,  -12,  9,     6,     1
[5]   -576,   30, 16,    30,     8,    1
[6]    105,   60, 25,   140,    57,   10,    1
[7]   5760, -140, 36,   630,   384,   90,   12,   1
[8] -13167, -280, 49,  2772,  2505,  772,  129,  14,  1
[9] -30400,  630, 64, 12012, 16008, 6430, 1328, 174, 16, 1 (End)

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..5 give A331551, A331552, A000290(n+1), A002457, A108666(n+1), A331323.
T(n,n+3) gives A331512.
Cf. A307883, A331513, A331514.

T := (n, k) -> (n + 1)^2*hypergeom([-n, -n], [2], k - 2):
seq(lprint(seq(simplify(T(n,k)), k=0..7)), n=0..6) # Peter Luschny, Jan 20 2020

T[n_, k_] := (n + 1)^2 * HypergeometricPFQ[{-n, -n}, {2}, k - 2];  Table[Table[T[n, k - n], {n, 0, k}], {k, 0, 9}] //Flatten (* Amiram Eldar, Jan 20 2020 *)

T(n,k) = Sum_{j=0..n} (k-3)^(n-j) * (n+j+1) * binomial(n,j) * binomial(n+j,j).
T(n,k) = Sum_{j=0..n} (k-2)^j * (j+1) * binomial(n+1,j+1)^2.
T(n,k) = (n + 1)^2*hypergeom([-n, -n], [2], k - 2). - Peter Luschny, Jan 20 2020
n * (2*n-1) * T(n,k) = 2 * (2 * (k-1) * n^2 - k + 2) * T(n-1,k) - (k-3)^2 * n * (2*n+1) * T(n-2,k) for n>1. - Seiichi Manyama, Jan 25 2020

cp's OEIS Frontend

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.

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A331511 Square array T(n,k), n >= 0, k >= 0, read by descending antidiagonals, where column k is the expansion of (1 - (k-3)x)/(1 - 2(k-1)x + ((k-3)x)^2)^(3/2).

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cp's OEIS Frontend

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.

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Magma

Maple

Mathematica

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A331511 Square array T(n,k), n >= 0, k >= 0, read by descending antidiagonals, where column k is the expansion of (1 - (k-3)*x)/(1 - 2*(k-1)*x + ((k-3)*x)^2)^(3/2).

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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.

A331511 Square array T(n,k), n >= 0, k >= 0, read by descending antidiagonals, where column k is the expansion of (1 - (k-3)x)/(1 - 2(k-1)x + ((k-3)x)^2)^(3/2).