A099577 -id:A099577 - 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

A099574 Diagonal sums of triangle A099573.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 9, 11, 18, 23, 37, 48, 74, 97, 147, 195, 290, 387, 568, 763, 1108, 1495, 2152, 2915, 4167, 5662, 8047, 10962, 15506, 21168, 29825, 40787, 57280, 78448, 109870, 150657, 210521, 288969, 403020, 553677, 770963, 1059932, 1473898

Offset: 0

Paul Barry, Oct 23 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A003269, A079977, A099573, A099577, A103609.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^4)/((1-x^2-x^4)*(1-x-x^4)) )); // G. C. Greubel, Jul 25 2022

a[n_]:= a[n]= Sum[Binomial[n-k-j, j], {k,0,Floor[n/2]}, {j,0,Floor[k/2]}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Jul 25 2022 *)

@CachedFunction
def A099574(n): return sum(sum(binomial(n-k-j, j) for j in (0..(k//2))) for k in (0..(n//2)))
[A099574(n) for n in (0..40)] # G. C. Greubel, Jul 25 2022

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..floor(k/2)} binomial(n-k-j, j).
G.f.: (1-x)*(1+x)*(1+x^2) / ( (1-x-x^4)*(1-x^2-x^4) ). - R. J. Mathar, Nov 11 2014
From G. C. Greubel, Jul 25 2022: (Start)
a(n) = A003269(n+5) - A079977(n+3) - A079977(n+2).
a(n) = A003269(n+5) - A103609(n+5). (End)

A099576 Row sums of triangle A099575.

Original entry on oeis.org

1, 2, 6, 12, 35, 72, 210, 440, 1287, 2730, 8008, 17136, 50388, 108528, 319770, 692208, 2042975, 4440150, 13123110, 28614300, 84672315, 185122080, 548354040, 1201610592, 3562467300, 7821594872, 23206929840, 51037462560, 151532656696

Offset: 0

Paul Barry, Oct 23 2004

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A029907, A099575, A099577.

[(&+[Binomial(n+Floor(k/2)+1, Floor(k/2)+1)*(1+Floor(k/2))/(n+1): k in [0..n]]): n in [0..40]]; // G. C. Greubel, Jul 24 2022

seq(op([(1+n/(n+1))*binomial(3*n+1,n),2*binomial(3*n+3,n)]),n=0..20);

a[n_] := Sum[Binomial[n + j, j], {k, 0, n}, {j, 0, k/2}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 06 2018 *)
a[n_] := Binomial[2*n+2, n]*Hypergeometric2F1[-n, n+1, -2*n-2, -1]; Flatten[Table[a[n], {n, 0, 28}]] (* Detlef Meya, Dec 25 2023 *)

a(n) = sum(k=0, n, sum(j=0, floor(k/2), binomial(n+j, j))); \\ Andrew Howroyd, Feb 13 2018

[sum( binomial(n+(k//2)+1, (k//2)+1)*(1+(k//2))/(n+1) for k in (0..n) ) for n in (0..40)] # G. C. Greubel, Jul 24 2022

a(n) = Sum_{k=0..n} Sum_{j=0..floor(k/2)} binomial(n+j, j).
Conjecture: 4*n*(n-1)*(3*n+2)*(n+2)*a(n) - 36*(n-1)*(n+1)*a(n-1) - 3*n*(3*n+5)*(3*n-1)*(3*n-2)*a(n-2) = 0. - R. J. Mathar, Nov 28 2014
From Robert Israel, May 08 2018: (Start)
a(2*n) = (1+n/(n+1))*binomial(3*n+1,n).
a(2*n+1) = 2*binomial(3*n+3,n).
The conjecture follows from this. (End)
a(n) = (1/(n+1))*Sum_{k=0..n} binomial(n + floor(k/2) + 1, floor(k/2) + 1)*(1 + floor(k/2)). - G. C. Greubel, Jul 24 2022
a(n) = binomial(2*n+2, n)*hypergeom([-n, n+1], [-2*n-2], -1). - Detlef Meya, Dec 25 2023

cp's OEIS Frontend

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

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A099574 Diagonal sums of triangle A099573.

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A099576 Row sums of triangle A099575.

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