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
Examples
Triangle begins:
1;
1, 2;
3, 8, 6;
17, 54, 60, 24;
155, 556, 762, 480, 120;
...
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.
Crossrefs
Programs
-
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
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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 *)
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
Extensions
More terms from David W. Wilson, Jan 12 2001
Comments