A261642 -id:A261642 - cp's OEIS frontend

A103244 Unreduced numerators of the elements T(n,k)/(n-k)!, read by rows, of the triangular matrix P^-1, which is the inverse of the matrix defined by P(n,k) = (-k^2-k)^(n-k)/(n-k)! for n >= k >= 1.

1, 2, 1, 20, 6, 1, 512, 108, 12, 1, 25392, 4104, 336, 20, 1, 2093472, 273456, 17568, 800, 30, 1, 260555392, 28515456, 1500288, 54800, 1620, 42, 1, 45819233280, 4311418752, 191549952, 5808000, 140400, 2940, 56, 1, 10849051434240, 894918533760, 34352605440, 887256000, 18033840, 313992, 4928, 72, 1

Offset: 1

Paul D. Hanna, Feb 02 2005

Define triangular matrix P by P(n,k) = (-k^2-k)^(n-k)/(n-k)!, then M = P*D*P^-1 = A103238 satisfies: M^2 + M = SHIFTUP(M) where D is the diagonal matrix consisting of {1,2,3,...}. The operation SHIFTUP(M) shifts each column of M up 1 row.

First column is A103353.

			This triangle begins:
1;
2, 1;
20, 6, 1;
512, 108, 12, 1;
25392, 4104, 336, 20, 1;
2093472, 273456, 17568, 800, 30, 1;
260555392, 28515456, 1500288, 54800, 1620, 42, 1;
45819233280, 4311418752, 191549952, 5808000, 140400, 2940, 56, 1;
10849051434240, 894918533760, 34352605440, 887256000, 18033840, 313992, 4928, 72, 1; ...
Rows of unreduced fractions T(n,k)/(n-k)! begin:
[1/0!],
[2/1!, 1/0!],
[20/2!, 6/1!, 1/0!],
[512/3!, 108/2!, 12/1!, 1/0!],
[25392/4!, 4104/3!, 336/2!, 20/1!, 1/0!],
[2093472/5!, 273456/4!, 17568/3!, 800/2!, 30/1!, 1/0!],...
forming the inverse of matrix P where P(n,k) = (-1)^(n-k)*(k^2+k)^(n-k)/(n-k)!:
[1/0!],
[ -2/1!, 1/0!],
[4/2!, -6/1!, 1/0!],
[ -8/3!, 36/2!, -12/1!, 1/0!],
[16/4!, -216/3!, 144/2!, -20/1!, 1/0!], ...

Cf. A261642, A103238, A103353.

nmax = 9;
P = Table[If[n >= k, (-k^2-k)^(n-k)/(n-k)!, 0], {n, 1, nmax}, {k, 1, nmax}] // Inverse;
T[n_, k_] := If[n < k || k < 1, 0, P[[n, k]]*(n - k)!];
Table[T[n, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 09 2018, from PARI *)

{T(n,k)=local(P);if(n>=k&k>=1, P=matrix(n,n,r,c,if(r>=c,(-c^2-c)^(r-c)/(r-c)!))); return(if(n

For n > k >= 1: 0 = Sum_{m=k..n} C(n-k, m-k)*(-m^2-m)^(n-m)*T(m, k).

For n > k >= 1: 0 = Sum_{j=k..n} C(n-k, j-k)*(-k^2-k)^(j-k)*T(n, j).

A261643 a(n) = Sum_{k=1..n} (k^2 + k)^(n-k).

Original entry on oeis.org

1, 3, 11, 57, 397, 3487, 37519, 484437, 7353473, 129104523, 2589967603, 58757627185, 1493762354293, 42223299711159, 1318186323111959, 45185985199663629, 1691822823829309801, 68865092213424362659, 3034735030143197197435, 144238580771432519823465, 7368717925255301486594525

Offset: 1

Paul D. Hanna, Aug 27 2015

nonn

Row sums of triangle A261642.

			Initial terms begin:
a(1) = 2^0 = 1;
a(2) = 2^1 + 6^0 = 3;
a(3) = 2^2 + 6^1 + 12^0 = 11;
a(4) = 2^3 + 6^2 + 12^1 + 20^0 = 57;
a(5) = 2^4 + 6^3 + 12^2 + 20^1 + 30^0 = 397;
a(6) = 2^5 + 6^4 + 12^3 + 20^2 + 30^1 + 42^0 = 3487; ...

Cf. A261642.

Table[Sum[(k^2+k)^(n-k),{k,n}],{n,30}] (* Harvey P. Dale, Aug 23 2021 *)

{a(n) = sum(k=1,n, (k + k^2)^(n-k))}
for(n=1,30,print1(a(n),", "))

a(n)^(1/n) ~ n^2/(exp(2)*LambertW(n)^2). - Vaclav Kotesovec, Aug 28 2015

cp's OEIS Frontend

A103244 Unreduced numerators of the elements T(n,k)/(n-k)!, read by rows, of the triangular matrix P^-1, which is the inverse of the matrix defined by P(n,k) = (-k^2-k)^(n-k)/(n-k)! for n >= k >= 1.

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