A036970 Triangle of coefficients of Gandhi polynomials.

1, 1, 2, 3, 8, 6, 17, 54, 60, 24, 155, 556, 762, 480, 120, 2073, 8146, 12840, 10248, 4200, 720, 38227, 161424, 282078, 263040, 139440, 40320, 5040, 929569, 4163438, 7886580, 8240952, 5170800, 1965600, 423360, 40320, 28820619, 135634292

Offset

1,3

Comments

Another version of triangle T(n,k), 0 <= k <= n, read by rows; given by [0, 1, 2, 4, 6, 9, 12, 16, 20, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, ...] = 1; 0, 1; 0, 1, 2; 0, 3, 8, 6; 0, 17, 54, 60, 24; ... where DELTA is the operator defined in A084938. - Philippe Deléham, Jun 07 2004

Links

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)

Peter Bala, The Gandhi polynomials as hypergeometric series

Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014.

R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015) # 15.3.2.

W. D. Cairns, Certain properties of binomial coefficients, Bull. Amer. Math. Soc. 26 (1920), 160-164. See p. 163 for a signed version.

Dominique Dumont, Sur une conjecture de Gandhi concernant les nombres de Genocchi, Discrete Mathematics 1 (1972) 321-327.

Dominique Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.

D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)

Marc Joye, Pascal Paillier and Berry Schoenmakers, On Second-Order Differential Power Analysis, in Cryptographic Hardware and Embedded Systems-CHES 2005, editors: Josyula R. Rao and Berk Sunar, Lecture Notes in Computer Science 3659 (2005) 293-308, Springer-Verlag.

Arthur Randrianarivony and Jiang Zeng, Une famille de polynomes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26.

Hans J. H. Tuenter, Walking into an absolute sum, arXiv:math/0606080 [math.NT], 2006. Published version on Walking into an absolute sum, The Fibonacci Quarterly, 40(2):175-180, May 2002.

Formula

Let B(X, n) = X^2 (B(X+1, n-1) - B(X, n-1)), B(X, 1) = X^2; then the (i, j)-th entry in the table is the coefficient of X^(1+j) in B(X, i). - Mike Domaratzki (mdomaratzki(AT)alumni.uwaterloo.ca), Nov 17 2001

From Gary W. Adamson, Jul 19 2011: (Start)

n-th row = top row of M^(n-1), M = an infinite square matrix in which the first "1" and right border of 1's of Pascal's triangle are deleted, as follows:

1, 2, 0, 0, 0, 0, ...

1, 3, 3, 0, 0, 0, ...

1, 4, 6, 4, 0, 0, ...

1, 5, 10, 10, 5, 0, ...

1, 6, 15, 20, 15, 6, ...

...

(End)

Let G(n,x) = (-1)^(n+1)*B(-x,n). Then G(n,x) = (2*x/(x+1))*( 1 + 2^(2*n+1)*(x-1)/(x+2) + 3^(2*n+1)*(x-1)*(x-2)/((x+2)*(x+3)) + ... ). Cf. A083061. - Peter Bala, Feb 04 2019

Example

Triangle begins:
    1;
    1,   2;
    3,   8,   6;
   17,  54,  60,  24;
  155, 556, 762, 480, 120;
  ...

Maple

B[1]:= X -> X^2:
for n from 2 to 12 do B[n]:= unapply(expand(X^2*(B[n-1](X+1)-B[n-1](X))), X) od:
seq(seq(coeff(B[i](X), X, 1+j), j=1..i), i=1..12); # Robert Israel, Apr 21 2016

Mathematica

B[1][X_] = X^2;
B[n_][X_] := B[n][X] = X^2*(B[n-1][X+1] - B[n-1][X]) // Simplify;
Table[Coefficient[B[i][X], X, j+1], {i, 1, 12}, {j, 1, i}] // Flatten (* Jean-François Alcover, Sep 19 2018, from Maple *)

Crossrefs

First 2 columns are Genocchi numbers A001469, A005440, row sums are also A001469.

Cf. A083061, A272378, A272379, A272380.

Sequence in context: A120390 A109230 A262992 * A110144 A327353 A270230

Adjacent sequences: A036967 A036968 A036969 * A036971 A036972 A036973

Keyword

tabl,nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson, Jan 12 2001

Status

approved