A102317 -id:A102317 - cp's OEIS frontend

A102316 Triangle, read by rows, where T(n,k) = T(n,k-1) + (k+1)*T(n-1,k) for n>k>0, T(n,0)=1 and T(n,n) = T(n,n-1) for n>=0.

1, 1, 1, 1, 3, 3, 1, 7, 16, 16, 1, 15, 63, 127, 127, 1, 31, 220, 728, 1363, 1363, 1, 63, 723, 3635, 10450, 18628, 18628, 1, 127, 2296, 16836, 69086, 180854, 311250, 311250, 1, 255, 7143, 74487, 419917, 1505041, 3683791, 6173791, 6173791, 1, 511, 21940

Offset: 0

Paul D. Hanna, Jan 04 2005

Main diagonal is A082161 (with offset). Row sums give A102317.

T(n,k) = number of column-marked subdiagonal paths of steps east (1,0) and north (0,1) from the origin to (n,k). Subdiagonal means that the path never rises above the diagonal line y=x and column-marked means that for 1 <= i <= n, one unit square directly below the i-th east step and above the line y=-1 is marked. - David Callan, Feb 04 2006

			T(5,2) = 220 = 1*1 + 2*15 + 3*63 = 1*T(4,0) + 2*T(4,1) + 3*T(4,2).
T(5,2) = 220 = 31 + 3*63 = T(5,1) + (2+1)*T(4,2).
T(5,3) = 728 = 220 + 4*127 = T(5,2) + (3+1)*T(4,3).
Rows begin:
[1],
[1,1],
[1,3,3],
[1,7,16,16],
[1,15,63,127,127],
[1,31,220,728,1363,1363],
[1,63,723,3635,10450,18628,18628],
[1,127,2296,16836,69086,180854,311250,311250],
[1,255,7143,74487,419917,1505041,3683791,6173791,6173791],...

Cf. A102086, A082161, A102317.

PARI
```
T(n,k)=if(n
				
```

T(n, k) = Sum_{j=0..k} (j+1)*T(n-1, j) for n>k>0, T(n, 0)=1 for n>=0. T(n, n) = A082161(n) for n>0; A082161(n+1) = Sum_{k=0..n} (k+1)*T(n, k).

A102400 Triangle, read by rows, where T(n,k) = Sum_{j=0..k} T(n-1,j)(j+1)[(k+1)(k+2)/2 - j(j+1)/2] for n>k>0, with T(0,0)=1 and T(n,n) = T(n,n-1) for n>0.

Original entry on oeis.org

1, 1, 1, 1, 7, 7, 1, 31, 139, 139, 1, 127, 1567, 5711, 5711, 1, 511, 15379, 126579, 408354, 408354, 1, 2047, 143527, 2357431, 15333661, 45605881, 45605881, 1, 8191, 1312219, 40769819, 473433344, 2634441290, 7390305396, 7390305396, 1, 32767

Offset: 0

Paul D. Hanna, Jan 06 2005

Main diagonal is A082162 (with offset). This sequence is derived from column 0 of A102098.

			T(4,2) = 1567 = 1*6 + 31*10 + 139*9
= T(3,0)*R(0,2) + T(3,1)*R(1,2) + T(3,2)*R(2,2).
Rows begin:
[1],
[1,1],
[1,7,7],
[1,31,139,139],
[1,127,1567,5711,5711],
[1,511,15379,126579,408354,408354],
[1,2047,143527,2357431,15333661,45605881,45605881],...
where the transpose of the recurrence coefficients given by
[R^t](n,k) = (k+1)*((n+1)*(n+2)/2 - k*(k+1)/2) form triangle:
[1],
[3,4],
[6,10,9],
[10,18,21,16],
[15,28,36,36,25],...
which equals the matrix square of the triangle:
[1],
[1,2],
[1,2,3],
[1,2,3,4],
[1,2,3,4,5],...

Cf. A082162, A102098, A102317.

T[n_, k_] := T[n, k] = If[nJean-François Alcover, Dec 15 2014, after PARI *)

PARI
```
{T(n,k)=if(n
				
```

cp's OEIS Frontend

A102316 Triangle, read by rows, where T(n,k) = T(n,k-1) + (k+1)*T(n-1,k) for n>k>0, T(n,0)=1 and T(n,n) = T(n,n-1) for n>=0.

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A102400 Triangle, read by rows, where T(n,k) = Sum_{j=0..k} T(n-1,j)(j+1)[(k+1)(k+2)/2 - j(j+1)/2] for n>k>0, with T(0,0)=1 and T(n,n) = T(n,n-1) for n>0.

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cp's OEIS Frontend

A102316 Triangle, read by rows, where T(n,k) = T(n,k-1) + (k+1)*T(n-1,k) for n>k>0, T(n,0)=1 and T(n,n) = T(n,n-1) for n>=0.

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Author

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Comments

Examples

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Programs

PARI

Formula

A102400 Triangle, read by rows, where T(n,k) = Sum_{j=0..k} T(n-1,j)*(j+1)*[(k+1)*(k+2)/2 - j*(j+1)/2] for n>k>0, with T(0,0)=1 and T(n,n) = T(n,n-1) for n>0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A102400 Triangle, read by rows, where T(n,k) = Sum_{j=0..k} T(n-1,j)(j+1)[(k+1)(k+2)/2 - j(j+1)/2] for n>k>0, with T(0,0)=1 and T(n,n) = T(n,n-1) for n>0.