A101494 - OEIS

%I #35 Feb 25 2021 10:55:54

%S 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,

%T 253,162,73,26,7,1,1,733,948,643,302,111,34,8,1,1,2781,3817,2724,1337,

%U 506,159,43,9,1,1,11378,16433,12259,6266,2457,788,218,53,10,1,1,49864,75295

%N 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.

%C Column 0 equals row sums (A026898) shift right.

%C 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

%H Muniru A Asiru, <a href="/A101494/b101494.txt">Rows n=0..150 of triangle, flattened</a>

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

%F T(n,k) = Sum_{j=0..n-k} binomial(j+k,j)*(n-k-j)^j. - _Vladeta Jovovic_, Sep 07 2006

%F 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

%F From _Mathew Englander_, Feb 25 2021: (Start)

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

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

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

%e 4th row sum = 23 = (5-0)^0+(5-1)^1+(5-2)^2+(5-3)^3+(5-4)^4.

%e 5th row sum = 66 = (6-0)^0+(6-1)^1+(6-2)^2+(6-3)^3+(6-4)^4+(6-5)^5.

%e T(6,0) = 66 = 1*23 + 1*23 + 1*13 + 1*5 + 1*1 + 1*1.

%e T(6,1) = 73 = 1*23 + 2*13 + 3*5 + 4*1 + 5*1.

%e T(6,2) = 44 = 1*13 + 3*5 + 6*1 + 10*1.

%e Rows begin:

%e 1;

%e 1, 1;

%e 2, 1, 1;

%e 4, 3, 1, 1;

%e 9, 8, 4, 1, 1;

%e 23, 23, 13, 5, 1, 1;

%e 66, 73, 44, 19, 6, 1, 1;

%e 210, 253, 162, 73, 26, 7, 1, 1;

%e 733, 948, 643, 302, 111, 34, 8, 1, 1;

%e 2781, 3817, 2724, 1337, 506, 159, 43, 9, 1, 1;

%e 11378, 16433, 12259, 6266, 2457, 788, 218, 53, 10, 1, 1;

%e 49864, 75295, 58423, 30953, 12558, 4147, 1163, 289, 64, 11, 1, 1;

%e 232769, 365600, 293902, 160823, 67259, 22878, 6574, 1647, 373, 76, 12, 1, 1; ...

%o (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));))

%o (PARI) T(n,k)=polcoeff(sum(m=0,n-k, x^m/(1-m*x +x*O(x^(n-k)))^(k+1)),n-k)

%o for(n=0,12,for(k=0,n,print1(T(n,k),", "));print()) \\ _Paul D. Hanna_, Mar 06 2013

%o (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

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

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Jan 21 2005