A099573 -id:A099573 - cp's OEIS frontend

A099574 Diagonal sums of triangle A099573.

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)

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

Original entry on oeis.org

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

cp's OEIS Frontend

A099574 Diagonal sums of triangle A099573.

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