A075915 -id:A075915 - cp's OEIS frontend

A075500 Stirling2 triangle with scaled diagonals (powers of 5).

1, 5, 1, 25, 15, 1, 125, 175, 30, 1, 625, 1875, 625, 50, 1, 3125, 19375, 11250, 1625, 75, 1, 15625, 196875, 188125, 43750, 3500, 105, 1, 78125, 1984375, 3018750, 1063125, 131250, 6650, 140, 1, 390625, 19921875

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.

The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(5*z) - 1)*x/5) - 1.

			[1]; [5,1]; [25,15,1]; ...; p(3,x) = x(25 + 15*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*     1
*     5       1
*    25      15       1
*   125     175      30       1
*   625    1875     625      50      1
*  3125   19375   11250    1625     75    1
* 15625  196875  188125   43750   3500  105   1
* 78125 1984375 3018750 1063125 131250 6650 140 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000351, A016164, A075911-A075915. Row sums are A005011(n-1).

Cf. A075499, A075501.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> 5^n, 9); # Peter Luschny, Jan 28 2016

Flatten[Table[5^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[5^#&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

for(n=1, 11, for(m=1, n, print1(5^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (5^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*5)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 5m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-5k*x), m >= 1.
E.g.f. for m-th column: (((exp(5x)-1)/5)^m)/m!, m >= 1.

A075914 Sixth column of triangle A075500.

Original entry on oeis.org

1, 105, 6650, 330750, 14266875, 560896875, 20682062500, 728227500000, 24779833203125, 821666548828125, 26708267167968750, 854772944238281250, 27023254648193359375, 846046877171630859375, 26282219820458984375000, 811330550012329101562500

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..5}(A075513(6,m)*exp(5*(m+1)*x))/5!.

Colin Barker, Table of n, a(n) for n = 0..675
Index entries for linear recurrences with constant coefficients, signature (105,-4375,91875,-1015000,5512500,-11250000).

Cf. A000351, A016164, A075911, A075912, A075913, A075915.

Table[5^(n-1) * (-1 + 5*2^(5+n) + 5*2^(11+2*n) - 10*3^(5+n) - 5^(6+n) + 6^(5+n))/24, {n, 0, 20}] (* Vaclav Kotesovec, Dec 12 2015 *)

Vec(1/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)*(1-30*x)) + O(x^30)) \\ Colin Barker, Dec 12 2015

a(n) = A075500(n+6, 6) = (5^n)*S2(n+6, 6) with S2(n, m) = A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..5}(A075513(6, m)*((m+1)*5)^n)/5!.
G.f.: 1/Product_{k=1..6}(1-5*k*x).
E.g.f.: (d^6/dx^6)((((exp(5*x)-1)/5)^6)/6!) = (-exp(5*x) + 160*exp(10*x) - 2430*exp(15*x) + 10240*exp(20*x) - 15625*exp(25*x) + 7776*exp(30*x))/5!.
G.f.: 1 / ((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)*(1-30*x)). - Colin Barker, Dec 12 2015

cp's OEIS Frontend

A075500 Stirling2 triangle with scaled diagonals (powers of 5).

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