A099576 -id:A099576 - cp's OEIS frontend

A099575 Number triangle T(n,k) = binomial(n + floor(k/2) + 1, n + 1), 0 <= k <= n.

1, 1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 6, 6, 21, 1, 1, 7, 7, 28, 28, 1, 1, 8, 8, 36, 36, 120, 1, 1, 9, 9, 45, 45, 165, 165, 1, 1, 10, 10, 55, 55, 220, 220, 715, 1, 1, 11, 11, 66, 66, 286, 286, 1001, 1001, 1, 1, 12, 12, 78, 78, 364, 364, 1365, 1365, 4368, 1, 1, 13, 13, 91, 91, 455, 455, 1820, 1820, 6188, 6188

Offset: 0

Paul Barry, Oct 23 2004

Original name was: "Number triangle T(n,k) = if(k<=n, Sum_{j=0..floor(k/2)} binomial(n+j,j), 0)."

			Rows start:
  1;
  1, 1;
  1, 1,  4;
  1, 1,  5,  5;
  1, 1,  6,  6, 21;
  1, 1,  7,  7, 28, 28;
  1, 1,  8,  8, 36, 36, 120;
  1, 1,  9,  9, 45, 45, 165, 165;
  1, 1, 10, 10, 55, 55, 220, 220, 715;

Robert Israel, Table of n, a(n) for n = 0..10010 (Rows 0..140, flattened)

Cf. A099573, A099576 (row sums), A099577 (diagonal sums), A099578 (main diagonal).

[Binomial(n+1+Floor(k/2), n+1): k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 24 2022

for n from 0 to 20 do seq(binomial(n+floor(k/2)+1,n+1),k=0..n) od; # Robert Israel, May 08 2018

Table[Binomial[n+Floor[k/2]+1, n+1], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 24 2022 *)

flatten([[binomial(n+(k//2)+1, n+1) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 24 2022

T(n, k) = binomial(n + floor(k/2) + 1, n + 1).
T(n, n) = A099578(n).
Sum_{k=0..n} T(n, k) = A099576(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A099577(n).

Definition simplified by Robert Israel, May 08 2018

A134565 Expansion of reversion of (x - 2*x^2) / (1 - x)^3.

Original entry on oeis.org

1, -1, 2, -3, 7, -12, 30, -55, 143, -273, 728, -1428, 3876, -7752, 21318, -43263, 120175, -246675, 690690, -1430715, 4032015, -8414640, 23841480, -50067108, 142498692, -300830572, 859515920, -1822766520, 5225264024, -11124755664, 31983672534, -68328754959

Offset: 1

Michael Somos, Nov 01 2007

sign

			G.f. = x - x^2 + 2*x^3 - 3*x^4 + 7*x^5 - 12*x^6 + 30*x^7 - 55*x^8 + 143*x^9 + ...

Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

Cf. A001764, A005161, A006013, A047749, A080956, A099576.

a[ n_] := With[ {m = Quotient[n, 2]}, If[n < 1, 0, -(-1)^n Binomial[n + m, n - m] / (2 m + 1)]]; (* Michael Somos, Oct 16 2015 *)
a[ n_] := If[n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[(x - 2 x^2) / (1 - x)^3, {x, 0, n}]], {x, 0, n}]]; (* Michael Somos, Oct 16 2015 *)
a[n_] := (-1)^(n-1)*Binomial[2*n, n-1]*Hypergeometric2F1[-n+1, n, -2*n, -1] / n; Flatten[Table[a[n], {n, 1, 32}]] (* Detlef Meya, Dec 26 2023 *)

{a(n) = my( m = n\2); if( n<1, 0, -(-1)^n * binomial( n + m, n - m) / (2 * m + 1))};

{a(n) = if( n<1, 0, polcoeff( serreverse( (x - 2 * x^2) / (1 - x)^3 + x * O(x^n) ), n))};

{a(n) = if( n<1, 0, polcoeff( 1 / ( 1 + 1 / serreverse( x - x^3 + x * O(x^n) )), n))};

Given g.f. A(x), then 1 = (1/A(x) + 1/A(-x)) / 2.
a(n) = -(-1)^n * binomial(n + m, n - m) / (2*m + 1) where m = floor(n/2) if n>0.
From Michael Somos, Apr 13 2012 (Start)
a(n) = -(-1)^n * A047749(n) unless n=0. a(2*n) = - A001764(n) unless n=0. a(2*n + 1) = A006013(n).
Reversion of A080956 with offset 1.
Hankel transform is A005161 omitting first 1.
n * a(n) = -(-1)^n * A099576(n-1). (End)
D-finite with recurrence +8*n*(n+1)*a(n) -36*n*(n-2)*a(n-1) +6*(-9*n^2+18*n-14)*a(n-2) +27*(3*n-7)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Sep 24 2021
a(n) = (-1)^(n-1)*binomial(2*n, n-1)*hypergeom([-n+1, n], [-2*n], -1) / n. - Detlef Meya, Dec 26 2023

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A099575 Number triangle T(n,k) = binomial(n + floor(k/2) + 1, n + 1), 0 <= k <= n.

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A134565 Expansion of reversion of (x - 2*x^2) / (1 - x)^3.

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