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A062253 2nd level triangle related to Eulerian numbers and binomial transforms (triangle of Eulerian numbers is first level and triangle with Z(0,0)=1 and Z(n,k)=0 otherwise is 0th level).

%I A062253 #11 Jan 27 2025 06:51:40
%S A062253 1,3,0,7,4,0,15,30,5,0,31,146,91,6,0,63,588,868,238,7,0,127,2136,6126,
%T A062253 4096,575,8,0,255,7290,36375,47400,16929,1326,9,0,511,23902,193533,
%U A062253 434494,306793,64362,2971,10,0,1023,76296,956054,3421902,4169418,1765940,231462,6534,11,0
%N A062253 2nd level triangle related to Eulerian numbers and binomial transforms (triangle of Eulerian numbers is first level and triangle with Z(0,0)=1 and Z(n,k)=0 otherwise is 0th level).
%C A062253 Binomial transform of n^2*k^n is ((kn)^2 + kn)*(k + 1)^(n - 2); of n^3*k^n is ((kn)^3 + 3(kn)^2 + (1 - k)(kn))*(k + 1)^(n - 3); of n^4*k^n is ((kn)^4 + 6(kn)^3 + (7 - 4k)(kn)^2 + (1 - 4k + k^2)(kn))*(k + 1)^(n - 4); of n^5*k^n is ((kn)^5 + 10(kn)^4 + (25 - 10k)(kn)^3 + (15 - 30k + 5k^2)(kn)^2 + (1 - 11k + 11k^2 - k^3)(kn))*(k + 1)^(n - 5); of n^6*k^n is ((kn)^6 + 15(kn)^5 + (65 - 20k)(kn)^4 + (90 - 120k + 15k^2)(kn)^3 + (31 - 146k + 91k^2 - 6k^3)(kn)^2 + (1 - 26k + 66k^2 - 26k^3 + k^4)(kn))*(k + 1)^(n - 6). This sequence gives the (unsigned) polynomial coefficients of (kn)^2.
%F A062253 A(n, k) = (k+2)*A(n-1, k)+(n-k)*A(n-1, k-1)+E(n, k) where E(n, k)=(k+1)*E(n-1, k)+(n-k)*E(n-1, k-1) and E(0, 0)=1 is a triangle of Eulerian numbers, essentially A008292.
%e A062253 Rows start:
%e A062253  (1),
%e A062253  (3,0),
%e A062253  (7,4,0),
%e A062253  (15,30,5,0),
%e A062253  ...
%o A062253 (PARI) E(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, (k+1)*E(n-1, k)+(n-k)*E(n-1, k-1)));
%o A062253 A2(n, k) = if ((n<0) || (k<0), 0, (k+2)*A2(n-1, k)+(n-k)*A2(n-1, k-1)+E(n, k));
%o A062253 row2(n) = vector(n+1, k, A2(n,k-1)); \\ _Michel Marcus_, Jan 27 2025
%Y A062253 First column is A000225. Diagonals include A000007, A009056. Row sums are A000254. Taking all the levels together to create a pyramid, one face would be A010054 as a triangle with a parallel face which is Pascal's triangle (A007318) with two columns removed, another face would be a triangle of Stirling numbers of the second kind (A008277) and a third face would be A000007 as a triangle, with a triangle of Eulerian numbers (A008292), A062253, A062254 and A062255 as faces parallel to it. The row sums of this last group would provide a triangle of unsigned Stirling numbers of the first kind (A008275).
%K A062253 nonn,tabl
%O A062253 0,2
%A A062253 _Henry Bottomley_, Jun 14 2001
%E A062253 More terms from _Michel Marcus_, Jan 27 2025