A101494 Triangle, read by rows, where T(n,k) = Sum_{j=0..n-k-1} C(j+k,j)*T(n-1,j+k) for n>k>=0 with T(n,n)=1.

1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 8, 4, 1, 1, 23, 23, 13, 5, 1, 1, 66, 73, 44, 19, 6, 1, 1, 210, 253, 162, 73, 26, 7, 1, 1, 733, 948, 643, 302, 111, 34, 8, 1, 1, 2781, 3817, 2724, 1337, 506, 159, 43, 9, 1, 1, 11378, 16433, 12259, 6266, 2457, 788, 218, 53, 10, 1, 1, 49864, 75295

Offset

0,4

Comments

Column 0 equals row sums (A026898) shift right.

T(n,k) is the number of m-tuples of nonnegative integers satisfying these two criteria: (i) there are exactly k 0’s, and (ii) the remaining m-k elements are positive integers less than or equal to n-m. - Mathew Englander, Feb 25 2021

Links

Muniru A Asiru, Rows n=0..150 of triangle, flattened

Formula

T(n,0) = A026898(n-1).

T(n,k) = Sum_{j=0..n-k} binomial(j+k,j)*(n-k-j)^j. - Vladeta Jovovic, Sep 07 2006

G.f.: A(x,y) = Sum_{n>=0} Sum_{k>=0} x^(n+k)*y^k / (1 - n*x)^(k+1). - Paul D. Hanna, Mar 06 2013

From Mathew Englander, Feb 25 2021: (Start)

G.f. of row n: Sum_{i=0..n} (x+n-i)^i.

T(n,k) = Sum_{j=k..n} A089246(j,k).

Antidiagonal sums: Sum_{j = 0..n} Sum_{i = j..floor((n+j)/2)} binomial(i,j)*(n+j-2*i)^j. (End)

Example

4th row sum = 23 = (5-0)^0+(5-1)^1+(5-2)^2+(5-3)^3+(5-4)^4.
5th row sum = 66 = (6-0)^0+(6-1)^1+(6-2)^2+(6-3)^3+(6-4)^4+(6-5)^5.
T(6,0) = 66 = 1*23 + 1*23 + 1*13 + 1*5 + 1*1 + 1*1.
T(6,1) = 73 = 1*23 + 2*13 + 3*5 + 4*1 + 5*1.
T(6,2) = 44 = 1*13 + 3*5 + 6*1 + 10*1.
Rows begin:
1;
1, 1;
2, 1, 1;
4, 3, 1, 1;
9, 8, 4, 1, 1;
23, 23, 13, 5, 1, 1;
66, 73, 44, 19, 6, 1, 1;
210, 253, 162, 73, 26, 7, 1, 1;
733, 948, 643, 302, 111, 34, 8, 1, 1;
2781, 3817, 2724, 1337, 506, 159, 43, 9, 1, 1;
11378, 16433, 12259, 6266, 2457, 788, 218, 53, 10, 1, 1;
49864, 75295, 58423, 30953, 12558, 4147, 1163, 289, 64, 11, 1, 1;
232769, 365600, 293902, 160823, 67259, 22878, 6574, 1647, 373, 76, 12, 1, 1; ...

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, if(n==k, 1, sum(j=0, n-k-1, binomial(j+k, j)*T(n-1, j+k)); ))
(PARI) T(n, k)=polcoeff(sum(m=0, n-k, x^m/(1-m*x +x*O(x^(n-k)))^(k+1)), n-k)
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print()) \\ Paul D. Hanna, Mar 06 2013
(GAP) Flat(List([0..10], n->List([0..n], k->Sum([0..n-k], j->Binomial(j+k, j)*(n-k-j)^j)))); # Muniru A Asiru, Mar 07 2019

Crossrefs

Cf. A101495, A026898, A089246 (first differences by column), A304357 (antidiagonal sums, empirically), A034856 (fourth diagonal).

Sequence in context: A105632 A091491 A117418 * A125781 A091150 A091351

Adjacent sequences: A101491 A101492 A101493 * A101495 A101496 A101497

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 21 2005

Status

approved