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A116924 Triangle T(n,k) = B(k,n) - B(k-1,n) where B(n,m) = Sum_{i=0..n} binomial(m,i) (3*i+1).

%I A116924 #11 Feb 13 2022 23:16:06
%S A116924 1,1,4,1,8,7,1,12,21,10,1,16,42,40,13,1,20,70,100,65,16,1,24,105,200,
%T A116924 195,96,19,1,28,147,350,455,336,133,22,1,32,196,560,910,896,532,176,
%U A116924 25,1,36,252,840,1638,2016,1596,792,225,28,1,40,315,1200
%N A116924 Triangle T(n,k) = B(k,n) - B(k-1,n) where B(n,m) = Sum_{i=0..n} binomial(m,i) (3*i+1).
%C A116924 The auxiliary array B(n,m) contains the binomial transform of the finite sequence of the first n+1 terms of 1, 4, 7, ... = A016777(.) in row n.
%C A116924 First few rows of the array B(n,m) are from row n=0 on:
%C A116924   1, 1,  1,  1,  1, ... (binomial transform of 1,0,0,0,...)
%C A116924   1, 5,  9, 13, 17, ... (binomial transform of 1,4,0,0,...)
%C A116924   1, 5, 16, 34, 59, ... (binomial transform of 1,4,7,0,...)
%C A116924   1, 5, 16, 44, 99, ... (binomial transform of 1,4,7,10,0,0,...)
%C A116924 The n-th row of triangle T contains the first differences of the n-th column of B.
%F A116924 Sum_{k=0..n} T(n,k) = A053220(n+1).
%e A116924 First few rows of the triangle T(n,k):
%e A116924   1;
%e A116924   1,  4;
%e A116924   1,  8,   7;
%e A116924   1, 12,  21,  10;
%e A116924   1, 16,  42,  40,  13;
%e A116924   1, 20,  70, 100,  65, 16;
%e A116924   1, 24, 105, 200, 195, 96, 19;
%e A116924   ...
%p A116924 B := proc(n,m) add(binomial(m,i)*(3*i+1),i=0..n) ; end proc:
%p A116924 A116924 := proc(n,k) if k = 0 then 1; else B(k,n)-B(k-1,n) ; end if; end proc:
%p A116924 seq(seq(A116924(n,k),k=0..n),n=0..15) ; # _R. J. Mathar_, Mar 27 2010
%Y A116924 Cf. A016777, A053220.
%K A116924 nonn,tabl,easy
%O A116924 0,3
%A A116924 _Gary W. Adamson_, Feb 26 2006
%E A116924 Replaced definition with formula, corrected from 5th row on by _R. J. Mathar_, Mar 27 2010