This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A036970 #87 Jan 05 2025 19:51:35
%S A036970 1,1,2,3,8,6,17,54,60,24,155,556,762,480,120,2073,8146,12840,10248,
%T A036970 4200,720,38227,161424,282078,263040,139440,40320,5040,929569,4163438,
%U A036970 7886580,8240952,5170800,1965600,423360,40320,28820619,135634292
%N A036970 Triangle of coefficients of Gandhi polynomials.
%C A036970 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
%H A036970 Robert Israel, <a href="/A036970/b036970.txt">Table of n, a(n) for n = 1..10011</a> (rows 1 to 141, flattened)
%H A036970 Peter Bala, <a href="/A036970/a036970.pdf">The Gandhi polynomials as hypergeometric series</a>
%H A036970 Richard P. Brent, <a href="http://arxiv.org/abs/1407.3533">Generalising Tuenter's binomial sums</a>, arXiv:1407.3533 [math.CO], 2014.
%H A036970 R. B. Brent, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Brent/brent5.html">Generalizing Tuenter's Binomial Sums</a>, J. Int. Seq. 18 (2015) # 15.3.2.
%H A036970 W. D. Cairns, <a href="http://dx.doi.org/10.1090/S0002-9904-1920-03278-7">Certain properties of binomial coefficients</a>, Bull. Amer. Math. Soc. 26 (1920), 160-164. See p. 163 for a signed version.
%H A036970 Dominique Dumont, <a href="http://dx.doi.org/10.1016/0012-365X(72)90039-8">Sur une conjecture de Gandhi concernant les nombres de Genocchi</a>, Discrete Mathematics 1 (1972) 321-327.
%H A036970 Dominique Dumont, <a href="http://dx.doi.org/10.1215/S0012-7094-74-04134-9">Interprétations combinatoires des nombres de Genocchi</a>, Duke Math. J., 41 (1974), 305-318.
%H A036970 D. Dumont, <a href="/A001469/a001469_3.pdf">Interprétations combinatoires des nombres de Genocchi</a>, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)
%H A036970 Marc Joye, Pascal Paillier and Berry Schoenmakers, <a href="http://www.win.tue.nl/~berry/papers/ches05hodpa.pdf">On Second-Order Differential Power Analysis</a>, 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.
%H A036970 Arthur Randrianarivony and Jiang Zeng, <a href="http://dx.doi.org/10.1006/aama.1996.0001">Une famille de polynomes qui interpole plusieurs suites classiques de nombres</a>, Adv. Appl. Math. 17 (1996), 1-26.
%H A036970 Hans J. H. Tuenter, <a href="https://arxiv.org/abs/math/0606080">Walking into an absolute sum</a>, arXiv:math/0606080 [math.NT], 2006. Published version on <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/40-2/tuenter.pdf">Walking into an absolute sum</a>, The Fibonacci Quarterly, 40(2):175-180, May 2002.
%F A036970 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
%F A036970 From _Gary W. Adamson_, Jul 19 2011: (Start)
%F A036970 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:
%F A036970 1, 2, 0, 0, 0, 0, ...
%F A036970 1, 3, 3, 0, 0, 0, ...
%F A036970 1, 4, 6, 4, 0, 0, ...
%F A036970 1, 5, 10, 10, 5, 0, ...
%F A036970 1, 6, 15, 20, 15, 6, ...
%F A036970 ...
%F A036970 (End)
%F A036970 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
%e A036970 Triangle begins:
%e A036970 1;
%e A036970 1, 2;
%e A036970 3, 8, 6;
%e A036970 17, 54, 60, 24;
%e A036970 155, 556, 762, 480, 120;
%e A036970 ...
%p A036970 B[1]:= X -> X^2:
%p A036970 for n from 2 to 12 do B[n]:= unapply(expand(X^2*(B[n-1](X+1)-B[n-1](X))),X) od:
%p A036970 seq(seq(coeff(B[i](X),X,1+j),j=1..i),i=1..12); # _Robert Israel_, Apr 21 2016
%t A036970 B[1][X_] = X^2;
%t A036970 B[n_][X_] := B[n][X] = X^2*(B[n-1][X+1] - B[n-1][X]) // Simplify;
%t A036970 Table[Coefficient[B[i][X], X, j+1], {i, 1, 12}, {j, 1, i}] // Flatten (* _Jean-François Alcover_, Sep 19 2018, from Maple *)
%Y A036970 First 2 columns are Genocchi numbers A001469, A005440, row sums are also A001469.
%Y A036970 Cf. A083061, A272378, A272379, A272380.
%K A036970 tabl,nonn,easy
%O A036970 1,3
%A A036970 _N. J. A. Sloane_
%E A036970 More terms from _David W. Wilson_, Jan 12 2001