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%I #44 Jun 09 2023 15:41:50
%S 1,2,-1,5,-6,2,14,-28,20,-5,42,-120,135,-70,14,132,-495,770,-616,252,
%T -42,429,-2002,4004,-4368,2730,-924,132,1430,-8008,19656,-27300,23100,
%U -11880,3432,-429,4862,-31824,92820,-157080,168300,-116688,51051,-12870,1430
%N Coefficient triangle for certain polynomials N(2; n,x) (rising powers of x).
%C The g.f. for the sequence of column m of triangle A009766(n,m) (or Catalan A033184(n,n-m) diagonals) is N(2; m-1,x)*(x^m)/(1-x)^(m+1), m >= 1, with N(2; n,x) := sum(a(n,k)*x^k,k=0..n).
%C For k=0..1 the column sequences give A000108(n+1) (Catalan), -A002694. The row sums give A000012 (powers of 1) and (unsigned) A062992.
%C Another version of [1, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [0, -1, -1, -1, -1, -1, -1, -1, ...] = 1; 1, 0; 2, -1, 0; 5, -6, 2, 0; 14, -28, 20, -5, 0; 42, -120, 135, -70, 14, 0; ... where DELTA is Deléham's operator defined in A084938.
%C The positive triangle has T(n,k)=binomial(2n+2,n-k)*binomial(n+k,k)/(n+1). - _Paul Barry_, May 11 2005
%H C. A. Francisco, J. Mermin, and J. Schweig, <a href="http://www.math.okstate.edu/~jayjs/ppt.pdf">Catalan numbers, binary trees, and pointed pseudotriangulations</a>, 2013.
%H V. E. Hoggatt Jr. and Marjorie Bicknell-Johnson, <a href="https://fq.math.ca/Scanned/15-1/hoggatt3.pdf">Numerator Polynomial Coefficient Arrays for Catalan and Related Sequence Convolution Triangles</a>, The Fibonacci Quarterly 15 (1977) 30-34. [On p. 31, in the line n = 1, 14 is missing in S_1^4. - _Wolfdieter Lang_, Jan 20 2020 ]
%H Joseph T. Iosue, Adam Ehrenberg, Dominik Hangleiter, Abhinav Deshpande, and Alexey V. Gorshkov, <a href="https://arxiv.org/abs/2209.06838">Page curves and typical entanglement in linear optics</a>, arXiv:2209.06838 [quant-ph], 2022.
%H A. Lakshminarayan, Z. Puchala, and K. Zyczkowski, <a href="http://arxiv.org/abs/1407.1169">Diagonal unitary entangling gates and contradiagonal quantum states</a>, arXiv preprint arXiv:1407.1169 [quant-ph], 2014.
%H Jian Zhou, <a href="https://arxiv.org/abs/2108.10514">On Some Mathematics Related to the Interpolating Statistics</a>, arXiv:2108.10514 [math-ph], 2021.
%F a(n, k) := [x^k]N(2; n, x) with N(2; n, x)=(N(2; n-1, x)-A000108(n)*(1-x)^(n+1))/x, N(2; 0, x) := 1.
%F a(n, k)= a(n-1, k+1)+((-1)^k)*binomial(n+1, k+1)*binomial(2*n+1, n)/(2*n+1) if k=0, .., (n-2); a( n, k)= ((-1)^k)*binomial(n+1, k+1)*binomial(2*n+1, n)/(2*n+1) if k=(n-1) or n; else 0.
%F O.g.f. (with offset 1) is the series reversion w.r.t. x of x*(1+x*t)/(1+x)^2. If R(n,t) denotes the n-th row polynomial of this triangle then R(n,1-t) is the n-th row polynomial of A009766. Cf. A089434. - _Peter Bala_, Jul 15 2012
%e The triangle N2 = {a(n,k)} begins:
%e n\k 0 1 2 3 4 5 6 7 8 9
%e ----------------------------------------------------------------------------
%e 0: 1
%e 1: 2 -1
%e 2: 5 -6 2
%e 3: 14 -28 20 -5
%e 4: 42 -120 135 -70 14
%e 5: 132 -495 770 -616 252 -42
%e 6: 429 -2002 4004 -4368 2730 -924 132
%e 7: 1430 -8008 19656 -27300 23100 -11880 3432 -429
%e 8: 4862 -31824 92820 -157080 168300 -116688 51051 -12870 1430
%e 9: 16796 -125970 426360 -852720 1108536 -969969 570570 -217360 48620 -4862
%e ... formatted by _Wolfdieter Lang_, Jan 20 2020
%e N(2; 2, x)= 5 - 6*x + 2*x^2.
%t T[n_, k_] := 2 (-1)^k Binomial[2n+1, n] (n-k+1) Binomial[n+1, k]/((k+n+1)(k+n+2));
%t Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 19 2018 *)
%Y Cf. A000012, A000108, A002694, A009766, A033184, A062992, A084938, A089434.
%Y For an unsigned version see Borel's triangle, A234950.
%K sign,easy,tabl
%O 0,2
%A _Wolfdieter Lang_, Jul 12 2001
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