A155834 A triangle sequence of general recursive Sierpinski-Pascal minus general Narayana with adjusted n,m levels and zeros out:k=2; t(n,m)=Pascal(n,m,k-1)-Narayana(n-1,m-1,2*(k-1)).

1, 1, 6, 16, 6, 22, 127, 127, 22, 64, 701, 1436, 701, 64, 163, 3117, 11503, 11503, 3117, 163, 382, 12088, 74122, 131494, 74122, 12088, 382, 848, 42890, 413612, 1193930, 1193930, 413612, 42890, 848, 1816, 143562, 2094588, 9280734, 14992440, 9280734

Offset: 4

Author

Roger L. Bagula, Jan 28 2009

Keywords

• nonn
• tabl
• uned

Comments

Row sums are;

2, 28, 298, 2966, 29566, 304678, 3302560, 38033840, 467861040, 6159690808,

86763791762,...

This level is the Eulerian number level:

only the odd Narayana levels correspond to the recursive Sierpinski-Pascal levels.

Examples

			{1, 1},
{6, 16, 6},
{22, 127, 127, 22},
{64, 701, 1436, 701, 64},
{163, 3117, 11503, 11503, 3117, 163},
{382, 12088, 74122, 131494, 74122, 12088, 382},
{848, 42890, 413612, 1193930, 1193930, 413612, 42890, 848},
{1816, 143562, 2094588, 9280734, 14992440, 9280734, 2094588, 143562, 1816},
{3797, 462541, 9928140, 64761204, 158774838, 158774838, 64761204, 9928140, 462541, 3797},
{7814, 1453700, 44960878, 418557816, 1489425900, 2250878592, 1489425900, 418557816, 44960878, 1453700, 7814},
{15914, 4495909, 197226603, 2558716162, 12781854516, 27839586777, 27839586777, 12781854516, 2558716162, 197226603, 4495909, 15914}
		

Crossrefs

A001263, A056939.A056941, A142465, A142467

Programs

Mathematica

Clear[A, a0, b0, n, k, m, t, i];
A[n_, 1, m_] := 1; A[n_, n_, m_] := 1;
A[n_, k_, m_] := (m*n - m*k + 1)*A[n - 1, k - 1, m] + (m*k - (m - 1))*A[n - 1, k, m];
t[n_, m_, i_] = Product[Binomial[n + k, m + k]/Binomial[n - m + k, k], {k, 0, i}];
m = 2; a = Table[A[n, k, m - 1] - t[n - 1, k - 1, (2*m - 2)], {n, 4, 14}, { k, 2, n - 1}];
Flatten[a]

Formula

Pascal(n,m,k):

a(n,k,m)=(m*n - m*k + 1)*a(n - 1, k - 1, m) + (m*k - (m - 1))*a(n - 1, k, m);

Narayana(n,m,k):

y(n,m,k)=Product[Binomial[n + k, m + k]/Binomial[n - m + k, k], {k, 0, i}];

k=2;

t(n,m)=Pascal(n,m,k-1)-Narayana(n-1,m-1,2*(k-1)).