A157901 Triangle read by rows: A000012 * A157898.

1, 1, 1, 2, 2, 2, 2, 4, 4, 4, 3, 6, 10, 8, 8, 3, 9, 16, 24, 16, 16, 4, 12, 28, 40, 56, 32, 32, 4, 16, 40, 80, 96, 128, 64, 64, 5, 20, 60, 120, 216, 224, 288, 128, 128, 5, 25, 80, 200, 336, 560, 512, 640, 256, 256, 6, 30, 110, 280, 616, 896, 1408, 1152, 1408, 512, 512

Offset: 0

Author

Gary W. Adamson & Roger L. Bagula, Mar 08 2009

Keywords

• nonn
• tabl
• changed

Comments

Multiplication of the lower triangular matrix A157898 from the left by A000012 means: these are partial column sums of A157898.

Examples

			First few rows of the triangle, n>=0:
  1;
  1,  1;
  2,  2,  2;
  2,  4,  4,   4;
  3,  6, 10,   8,   8;
  3,  9, 16,  24,  16,  16;
  4, 12, 28,  40,  56,  32,  32;
  4, 16, 40,  80,  96, 128,  64,  64;
  5, 20, 60, 120, 216, 224, 288, 128, 128;
  5, 25, 80, 200, 336, 560, 512, 640, 256, 256;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000012, A105635, A157898.

Columns: A004526 (k=0), A002620 (k=1), A006584 (k=2), 4*A096338 (k=3), 8*A177747 (k=4), 16*A299337 (k=5), 32*A178440 (k=6).

Sums include: A105635(n+1) (row), A166486(n+1) (alternating sign diagonal), A232801(n+1) (diagonal).

Programs

Magma

A011782:= func< n | n eq 0 select 1 else 2^(n-1) >;
function t(n, k) // t = A059576
  if k eq 0 or k eq n then return A011782(n);
  else return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1);
  end if; return t;
end function;
A157898:= func< n, k | (&+[(-1)^(n-j)*Binomial(n, j)*t(j, k): j in [k..n]]) >;
A157071:= func< n,k | (&+[A157898(j+k,k): j in [0..n-k]]) >;
[A157071(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 27 2025

Mathematica

t[n_, k_]:= t[n,k]= If[k==0 || k==n, 2^(n-1) +Boole[n==0]/2, 2*t[n-1,k-1] +2*t[n-1,k] - (2 -Boole[n==2])*t[n-2,k-1]]; (* t = A059576 *)
A157898[n_, k_]:= A157898[n,k]= Sum[(-1)^(n-j)*Binomial[n,j]*t[j,k], {j,k,n}];
A157901[n_, k_]:= A157901[n,k]= Sum[A157898[j+k,k], {j,0,n-k}];
Table[A157901[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 27 2025 *)

SageMath

@CachedFunction
def t(n, k): # t = A059576
    if (k==0 or k==n): return bool(n==0)/2 + 2^(n-1) # A011782
    else: return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1)
def A157898(n, k): return sum((-1)^(n+k-j)*binomial(n, j+k)*t(j+k, k) for j in range(n-k+1))
def A157071(n,k): return sum(A157898(j+k,k) for j in range(n-k+1))
print(flatten([[A157071(n,k) for k in range(n+1)] for n in range(10)])) # G. C. Greubel, Aug 27 2025

Formula

T(n,k) = Sum_{j=0..n} A157898(j,k).

Extensions

Edited by the Associate Editors of the OEIS, Apr 10 2009

More terms from G. C. Greubel, Aug 27 2025