A062991 - OEIS

A062991

Coefficient triangle for certain polynomials N(2; n,x) (rising powers of x).

%I #50 Sep 27 2024 09:24:34

%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_{k=0..n} T(n,k)*x^k.

%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 is |T(n,k)| = binomial(2*n+2, n-k)*binomial(n+k, k)/(n+1). - _Paul Barry_, May 11 2005

%H G. C. Greubel, <a href="/A062991/b062991.txt">Rows n = 0..50 of the triangle, flattened</a>

%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 T(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 T(n, k) = T(n-1, k+1) + (-1)^k*binomial(n+1, k+1)*binomial(2*n+1, n)/(2*n+1) if k=0, .., (n-2); T(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

%F From _G. C. Greubel_, Sep 27 2024: (Start)

%F Sum_{k=0..n} T(n, k) = A000012(n).

%F Sum_{k=0..n} (-1)^k*T(n, k) = A064062(n+1).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = A000079(n).

%F Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A071356(n). (End)

%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 *)

%o (Magma)

%o A062991:= func< n,k | (-1)^k*Binomial(2*n+2,n-k)*Binomial(n+k,k)/(n+1) >;

%o [A062991(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 27 2024

%o (SageMath)

%o def A062991(n,k): return (-1)^k*binomial(2*n+2,n-k)*binomial(n+k,k)/(n+1)

%o flatten([[A062991(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Sep 27 2024

%Y Cf. A000108, A002694, A009766, A033184, A062992, A084938, A089434, A094385.

%Y For an unsigned version see Borel's triangle, A234950.

%Y Sums include: A000012 (row), A000079 (diagonal), A064062 (signed row), A071356 (signed diagonal).

%K sign,easy,tabl

%O 0,2

%A _Wolfdieter Lang_, Jul 12 2001