A163774 -id:A163774 - cp's OEIS frontend

A163771 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients (A002426) with the central binomial coefficients (A000984).

1, 1, 2, 3, 4, 6, 7, 10, 14, 20, 19, 26, 36, 50, 70, 51, 70, 96, 132, 182, 252, 141, 192, 262, 358, 490, 672, 924, 393, 534, 726, 988, 1346, 1836, 2508, 3432, 1107, 1500, 2034, 2760, 3748, 5094, 6930, 9438, 12870

Offset: 0

Peter Luschny, Aug 05 2009

Triangle read by rows. For n >= 0, k >= 0 let T(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i)$ where i$ denotes the swinging factorial of i (A056040).

This is also the square array of central binomial coefficients A000984 in column 0 and higher (first: A051924, second, etc.) differences in subsequent columns, read by antidiagonals. - M. F. Hasler, Nov 15 2019

			Triangle begins
    1;
    1,   2;
    3,   4,   6;
    7,  10,  14,  20;
   19,  26,  36,  50,  70;
   51,  70,  96, 132, 182, 252;
  141, 192, 262, 358, 490, 672, 924;
From _M. F. Hasler_, Nov 15 2019: (Start)
The square array having central binomial coefficients A000984 in column 0 and higher differences in subsequent columns (col. 1 = A051924) starts:
     1   1    3    7    19    51 ...
     2   4   10   26    70   192 ...
     6  14   36   96   262   726 ...
    20  50  132  358   988  2760 ...
    70 182  490 1346  3748 10540 ...
   252 672 1836 5094 14288 40404 ...
  (...)
Read by falling antidiagonals this yields the same sequence. (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.
M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

Row sums are A163774. Cf. A056040, A163650, A163771, A163772, A002426, A000984.

For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle.
a := n -> DiffTria(k->swing(2*k),n,true);

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

A264766 Irregular symmetric triangle of coefficients T(n,k) of the polynomials p(n,x) = Sum_{k=0..n} binomial(n+1,k)(1+x)^(2k)(-x)^(n-k) for 0 <= k <= 2n.

Original entry on oeis.org

1, 2, 3, 2, 3, 9, 13, 9, 3, 4, 18, 40, 51, 40, 18, 4, 5, 30, 90, 165, 201, 165, 90, 30, 5, 6, 45, 170, 405, 666, 783, 666, 405, 170, 45, 6, 7, 63, 287, 840, 1736, 2646, 3039, 2646, 1736, 840, 287, 63, 7, 8, 84, 448, 1554, 3864, 7224, 10424, 11763, 10424, 7224, 3864, 1554, 448, 84, 8, 9, 108, 660, 2646, 7686, 17010, 29520, 40851, 45481, 40851, 29520, 17010, 7686, 2646, 660, 108, 9, 10

Offset: 0

Werner Schulte, Nov 23 2015

			The irregular triangle T(n,k) begins:
n\k:  0   1    2     3     4     5      6      7      8     9    10  11  12
  0:  1
  1:  2   3    2
  2:  3   9   13     9     3
  3:  4  18   40    51    40    18      4
  4:  5  30   90   165   201   165     90     30      5
  5:  6  45  170   405   666   783    666    405    170    45     6
  6:  7  63  287   840  1736  2646   3039   2646   1736   840   287  63   7
  etc.
The polynomial corresponding to row 2 is p(2,x) = 3 + 9*x + 13*x^2 + 9*x^3 + 3*x^4.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000108, A000984, A001006, A002426, A003462, A005061, A027907, A058987, A163774, A260056.

T[n_, k_] := Sum[(-1)^j*Binomial[n + 1, j + 1]*Binomial[2*n - 2*j, k - j], {j, 0, n - Abs[k - n]}]; Table[T[n, k], {n,0,10}, {k,0,2*n}] // Flatten (* G. C. Greubel, Aug 12 2017 *)

T(n,k) = sum(j=0, n-abs(k-n), (-1)^j*binomial(n+1,j+1)*binomial(2*n-2*j,k-j));
tabf(nn) = for (n=0, nn, for (k=0, 2*n, print1(T(n, k), ", ");); print();); \\ Michel Marcus, Nov 24 2015

T(n,k) = Sum_{j=0..n-d} (-1)^j*binomial(n+1,j+1)*binomial(2*n-2*j,k-j) if d = 0 or better d = abs(k-n), and 0 <= k <= 2*n.
Recurrence: T(n,0) = n+1, and T(n,k) = 0 for k < 0 or k > 2*n, and T(n+1,k) = T(n,k-2) + T(n,k-1) + T(n,k) + binomial(2*n+2,k) for k > 0 and n >= 0.
T(n,k) = T(n,2*n-k) for 0 <= k <= 2*n.
p(n,x) = Sum_{k=0..2*n} T(n,k)*x^k = Sum_{k=0..n} (1+x)^(2*k)*(1+x+x^2)^(n-k) = Sum_{k=0..n} binomial(n+1,k)*(1+x+x^2)^k*x^(n-k) for n >= 0.
Recurrence: p(0,x) = 1, and p(n+1,x) = (1+x+x^2)*p(n,x)+(1+x)^(2*n+2), n >= 0.
T(n,n) = Sum_{j=0..n} (-1)^(n-j)*binomial(n+1,j)*binomial(2*j,j) = A000984(n+1)-A002426(n+1) for n >= 0 (see also A163774).
Sum_{n>=0} T(n,n)*x^(n+1) = 1/sqrt(1-4*x) - 1/sqrt(1-2*x-3*x^2) for abs(x) < 1/4.
T(n,n-1) = binomial(2*n+2,n) - A027907(n+1,n) for n > 0.
T(n+1,n)/(n+2) = A000108(n+2) - A001006(n+1) for n >= 0 (see also A058987).
Row sums: p(n,1) = A005061(n+1) for n >= 0.
Alternating row sums: p(n,-1) = 1 for n >= 0.
p(n,-2) = Sum_{k=0..2*n} T(n,k)*(-2)^k = A003462(n+1) for n >= 0.
T(n,k) = Sum_{j=0..k} (-1)^j*A260056(n,j)*binomial(2*n-j,k-j) for 0 <= k <= 2*n.
A260056(n,k) = Sum_{j=0..k} (-1)^j*T(n,j)*binomial(2*n-j,k-j) for 0 <= k <= 2*n.
p(n,-1-x) = Sum{k=0..2*n} A260056(n,k)*x^(2*n-k) for n >= 0.
p(n,-x/(1+x))*(1+x)^(2*n) = Sum_{k=0..2*n} A260056(n,k)*x^k for n >= 0.
Sum_{n>=0} p(n,x)*t^n = 1/((1-t*(1+x)^2)*(1-t*(1+x+x^2))).
p(n,x)*x = (1+x)^(2*n+2) - (1+x+x^2)^(n+1), n >= 0.
T(n,k) = binomial(2*n+2,k+1) - A027907(n+1,k+1) for 0 <= k <= 2*n.

cp's OEIS Frontend

A163771 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients (A002426) with the central binomial coefficients (A000984).

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A264766 Irregular symmetric triangle of coefficients T(n,k) of the polynomials p(n,x) = Sum_{k=0..n} binomial(n+1,k)(1+x)^(2k)(-x)^(n-k) for 0 <= k <= 2n.

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cp's OEIS Frontend

A163771 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients (A002426) with the central binomial coefficients (A000984).

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Maple

Mathematica

A264766 Irregular symmetric triangle of coefficients T(n,k) of the polynomials p(n,x) = Sum_{k=0..n} binomial(n+1,k)*(1+x)^(2*k)*(-x)^(n-k) for 0 <= k <= 2*n.

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A264766 Irregular symmetric triangle of coefficients T(n,k) of the polynomials p(n,x) = Sum_{k=0..n} binomial(n+1,k)(1+x)^(2k)(-x)^(n-k) for 0 <= k <= 2n.