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.

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

Author

Werner Schulte, Nov 23 2015

Keywords

• nonn
• easy
• tabf

Examples

			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.
		

Links

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

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

Formula

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.