cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A111672 Array T(n,k) = A153277(n-1,k) = A144150(n,k-1) read by downwards antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 12, 15, 1, 1, 5, 22, 60, 52, 1, 1, 6, 35, 154, 358, 203, 1, 1, 7, 51, 315, 1304, 2471, 877, 1, 1, 8, 70, 561, 3455, 12915, 19302, 4140, 1
Offset: 1

Views

Author

Gary W. Adamson, Aug 14 2005

Keywords

Comments

Column k is obtained by taking the k-th matrix power of the triangle A008277 and multiplying from the right with the column vector [1,0,0,0,....].

Examples

			The array starts
1,  1,   1,    1,    1,    1,  ...
1,  2,   3,    4,    5,    6,  ...
1,  5,  12,   22,   35,   51,  ...
1, 15,  60,  154,  315,  561,  ...
1, 52, 358, 1304, 3455, 7556,  ...

Links

Crossrefs

Cf. A008277, A144150, A153277.
Cf. A000326 (row 3), A005945 (row 4), A000110 (column 2), A000258 (column 3), A000307 (column 4), A000357 (column 5), A000405 (column 6), A111669 (column 7), A081624.

Extensions

a(44) and definition corrected by Georg Fischer, May 18 2022

A003659 Shifts left under Stirling2 transform.

Original entry on oeis.org

1, 1, 2, 6, 26, 152, 1144, 10742, 122772, 1673856, 26780972, 496090330, 10519217930, 252851833482, 6832018188414, 205985750827854, 6885220780488694, 253685194149119818, 10250343686634687424, 452108221967363310278, 21676762640915055856716
Offset: 1

Views

Author

N. J. A. Sloane, Mira Bernstein

Keywords

Comments

Apart from leading term, number of M-sequences from multicomplexes on at most 4 variables with no monomial of degree more than n+1.
Stirling2 transform of a(n) = [1, 1, 2, 6, 26, ...] is a(n+1) = [1, 2, 6, 26, ...].
Eigensequence of Stirling2 triangle A008277. - Philippe Deléham, Mar 23 2007

References

  • S. Linusson, The number of M-sequences and f-vectors, Combinatorica, 19 (1999), 255-266.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A048801.
Cf. A153277, A153278. - Jonathan Vos Post, Dec 22 2008

Programs

  • Maple
    stirtr:= proc(p)
           proc(n) add(p(k)*Stirling2(n,k), k=0..n) end
         end:
a:= proc(n) option remember; `if`(n<3, 1, aa(n-1)) end:
aa:= stirtr(a):
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 22 2012
  • Mathematica
    terms = 21; A[] = 0; Do[A[x] = Normal[Integrate[1 + A[Exp[x] - 1 + O[x]^(terms + 1)], x] + O[x]^(terms + 1)], terms];
CoefficientList[A[x], x]*Range[0, terms]! // Rest (* Jean-François Alcover, May 23 2012, updated Jan 12 2018 *)
  • PARI
    {a(n)=local(A, E); if(n<0, 0, A=O(x); E=exp(x+x*O(x^n))-1; for(m=1, n, A=intformal( subst( 1+A, x, E+x*O(x^m)))); n!*polcoeff(A, n))} /* Michael Somos, Mar 08 2004 */
  • PARI
    a_vector(n) = my(v=vector(n)); v[1]=1; for(i=1, n-1, v[i+1]=sum(j=1, i, stirling(i, j, 2)*v[j])); v; \\ Seiichi Manyama, Jun 24 2022

Formula

E.g.f. A(x) satisfies A(x)' = 1+A(exp(x)-1).
G.f. satisfies: Sum_{n>=1} a(n)*x^n = x * (1 + Sum_{n>=1} a(n) * x^n / Product_{j=1..n} (1 - j*x)). - Ilya Gutkovskiy, May 09 2019
a(1) = 1; a(n+1) = Sum_{k=1..n} Stirling2(n,k) * a(k). - Seiichi Manyama, Jun 24 2022

A153278 Array read by antidiagonals of higher order Fubini numbers.

Original entry on oeis.org

1, 1, 3, 1, 4, 13, 1, 5, 23, 75, 1, 6, 36, 175, 541, 1, 7, 52, 342, 1662, 4683, 1, 8, 71, 594, 4048, 18937, 47293, 1, 9, 93, 949, 8444, 57437, 251729, 545835, 1, 10, 118, 1425, 15775, 143783, 950512, 3824282, 7087261, 1, 11, 146, 2040, 27146, 313920, 2854261, 17975438, 65361237, 102247563
Offset: 1

Views

Author

Jonathan Vos Post, Dec 22 2008

Keywords

Examples

			The table on p.6 of Mezo begins:
===========================================================
F_p,n|n=1|n=2|n=3.|.n=4.|..n=5.|....n=6.|.....n=7.|comment
===========================================================
p=1..|.1.|.3.|.13.|..75.|..541.|...4683.|...47293.|.A000670
p=2..|.1.|.4.|.23.|.175.|.1662.|..18937.|..251729.|.A083355
p=3..|.1.|.5.|.36.|.342.|.4048.|..57437.|..950512.|.A099391
p=4..|.1.|.6.|.52.|.594.|.8444.|.143783.|.2854261.|.A363008
p=5..|.1.|.7.|.71.|.949.|15775.|.313920.|.7279795.|.A363009
===========================================================

Links

Crossrefs

Cf. A000670, A083355, A099391, A153277, A363008, A363009.

Programs

  • Maple
    f:= proc(n) option remember; local k; if n<=1 then 1 else
       add(binomial(n, k) *f(n-k), k=1..n) fi
    end:
stirtr:= proc(a) proc(n) option remember;
           add( a(k) *Stirling2(n,k), k=0..n)
         end end:
F:= (p,n)-> (stirtr@@(p-1))(f)(n):
seq(seq(F(d-n, n), n=1..d-1), d=1..13); # Alois P. Heinz, Feb 02 2009
  • Mathematica
    f[n_] := f[n] = If[n <= 1, 1, Sum[Binomial[n, k]*f[n-k], {k, 1, n}]];
stirtr[a_] := Module[{g}, g[n_] := g[n] = Sum[a[k]*StirlingS2[n, k], {k, 0, n}]; g];
F[p_, n_] := (Composition @@ Table[stirtr, {p-1}])[f][n];
Table[Table[F[d-n, n], {n, 1, d-1}], {d, 1, 13}] // Flatten (* Jean-François Alcover, Mar 30 2016, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Feb 02 2009
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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