A134141 -id:A134141 - cp's OEIS frontend

A132164 Row sums of triangle A134141 (S1p(7)).

1, 1, 8, 78, 918, 12846, 209616, 3909228, 81859548, 1897344828, 48135826656, 1325008302696, 39292978029768, 1247949491330088, 42236558731574208, 1516738194700667856, 57573649342673292816, 2302425590703685075728, 96720470167595138898048

Offset: 0

Wolfdieter Lang, Oct 12 2007

Alois P. Heinz, Table of n, a(n) for n = 0..409

Cf. A132165 (alternating row sum of A134141), A049428.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+5)!/6!*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017

a[n_]:=a[n]=If[n==0, 1, Sum[Binomial[n - 1, j - 1] (j + 5)!/6! a[n - j], {j, n}]]; Table[a[n], {n, 0, 25}] (* Indranil Ghosh, Aug 02 2017, after Maple code *)

a(n)= sum(A134141(n,m),m=1..n),n>=1.
E.g.f.: exp((1-(1-x)^6)/(6*(1-x)^6)). Cf. e.g.f. first column of A134141.
From Seiichi Manyama, Jan 18 2025: (Start)
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A005012(k).
a(n) = (1/exp(1/6)) * (-1)^n * n! * Sum_{k>=0} binomial(-6*k,n)/(6^k * k!). (End)

a(0)=1 prepended by Alois P. Heinz, Aug 01 2017

A132165 Alternating row sums of triangle A134141 (S1p(7)).

Original entry on oeis.org

1, 6, 36, 174, -174, -26076, -596616, -10334556, -143486244, -1157102136, 19891298736, 1443409042536, 52915941558936, 1542313273333104, 37123741709607456, 635436136388469744, -1073509544497887216, -801332217409556182176, -51221481393025808800704

Offset: 1

Wolfdieter Lang Oct 12 2007

Cf. A132164 (row sum of A134141).

a(n)= sum(A134141(n,m)*(-1)^(m-1),m=1..n),n>=1.

E.g.f.: 1 - exp(-(1-(1-x)^6)/(6*(1-x)^6)). Cf. e.g.f. first column of A134141.

A132166 A convolution triangle of numbers obtained from A036224.

Original entry on oeis.org

1, 21, 1, 336, 42, 1, 4536, 1113, 63, 1, 54432, 23184, 2331, 84, 1, 598752, 412272, 65205, 3990, 105, 1, 6158592, 6531840, 1518048, 139860, 6090, 126, 1, 60046272, 94618368, 30912840, 4010769, 256410, 8631, 147, 1, 560431872, 1274921856

Offset: 1

Wolfdieter Lang, Oct 12 2007

Signed version: (-1)^(n-m)*a(n, m) := s1(7; n,m).

a(n,m) := s1p(7; n,m), a member of a sequence of unsigned triangles including s1p(2; n,m)= A007318(n-1,m-1) (Pascal's triangle), A030523=s1p(3), A036068=s1p(4), A030526=s1p(5) and A030527=s1p(6).

			{1};{21,1};{336,42,1};{4536,1113,63,1};...; Row polynomial s(3,x)=336*x+42*x^2+x^3.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First ten rows.

Related triangle A134141 (S1p(7)).

Cf. A036224(n-1), n>=1 (first column). A132167 (row sums). A132168 (alternating row sums).

a(n, m) = 6*(6*m+n-1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n

G.f. for m-th column: ((1-(1-6*x)^6)/(36*(1-6*x)^6))^m.

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A132164 Row sums of triangle A134141 (S1p(7)).

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A132165 Alternating row sums of triangle A134141 (S1p(7)).

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A132166 A convolution triangle of numbers obtained from A036224.

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