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 1-8 of 8 results.

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

A213357 E.g.f. satisfies A(x) = 1 + (exp(x) - 1) * A(exp(x) - 1).

Original entry on oeis.org

1, 1, 3, 16, 133, 1561, 24374, 485640, 11969843, 356348290, 12572687675, 517644938724, 24553141710156, 1327223189312606, 81005220402829714, 5537660009982114858, 421050946315817655785, 35387457515051683169307, 3269500807582223015227780
Offset: 0

Views

Author

Michael Somos, Jun 09 2012

Keywords

Examples

			1 + x + 3*x^2 + 16*x^3 + 133*x^4 + 1561*x^5 + 24374*x^6 + 485640*x^7 + ...

Links

Crossrefs

Cf. A048801, A135750.

Programs

  • Mathematica
    nmax=20; b = ConstantArray[0,nmax+1]; b[[1]]=1; Do[b[[n+1]] = Sum[k*b[[k]]*StirlingS2[n, k],{k,1,n}],{n,1,nmax-1}]; b (* Vaclav Kotesovec, Mar 12 2014 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = subst( 1 + x * A, x, exp( x + x * (A - A)) - 1)); n! * polcoeff( A, n))}
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*stirling(i, j, 2)*v[j])); v; \\ Seiichi Manyama, Jun 04 2022

Formula

a(n) = Sum_{k=1..n} k * a(k-1) * Stirling2(n, k) if n>0.
A048801(n) = n * a(n-1) = Sum_{k=1..n} a(k) * Stirling1(n, k) if n>0.

A354729 E.g.f. A(x) satisfies A(x) = 1 + x * A(log(1+x)).

Original entry on oeis.org

1, 1, 2, 3, -4, -30, 234, 679, -35848, 305208, 6762360, -290545486, 2866197828, 186075548048, -10575881477630, 151622861284395, 14937532353298992, -1269964031741331704, 32904195657758601624, 2814524425307181390432, -395787864674458924551840
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2022

Keywords

Crossrefs

Cf. A048801, A354728, A354730.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*sum(j=0, i-1, stirling(i-1, j, 1)*v[j+1])); v;

Formula

a(0) = 1; a(n) = n * Sum_{k=0..n-1} Stirling1(n-1,k) * a(k).
a(n) = n * A354728(n-1) for n>0.

A354730 E.g.f. A(x) satisfies: A(x) = 1 + x * A(-log(1-x)).

Original entry on oeis.org

1, 1, 2, 9, 68, 750, 11214, 216559, 5217176, 152742528, 5324034480, 217322508194, 10248159667140, 551968543756448, 33627829222316770, 2298114390067518705, 174893648144384932176, 14727538317383970615352, 1364522959678851181512504
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2022

Keywords

Crossrefs

Cf. A048801, A135750, A354729.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*sum(j=0, i-1, abs(stirling(i-1, j, 1))*v[j+1])); v;

Formula

a(0) = 1; a(n) = n * Sum_{k=0..n-1} |Stirling1(n-1,k)| * a(k).
a(n) = n * A135750(n-1) for n>0.

A354574 E.g.f. A(x) satisfies A(x) = 1 + x * A(1 - exp(-x)).

Original entry on oeis.org

1, 1, 2, 3, -8, -65, 366, 4284, -71392, -377919, 28218760, -249587877, -14356069056, 587285561746, 153563287892, -954498079774950, 39921820513516256, 533333406684245239, -158979463609003391970, 8008135971419079188618, 190727236066813163686860
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2022

Keywords

Crossrefs

Cf. A048801, A353177, A354729, A354730.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*sum(j=0, i-1, (-1)^(i-j-1)*stirling(i-1, j, 2)*v[j+1])); v;

Formula

a(0) = 1; a(n) = n * Sum_{k=0..n-1} (-1)^(n-k-1) * Stirling2(n-1,k) * a(k).
a(n) = n * A353177(n-1) for n>0.

A355100 E.g.f. A(x) satisfies A(x) = 1 + 2 * x * A(exp(x) - 1).

Original entry on oeis.org

1, 2, 8, 60, 688, 11060, 234744, 6314196, 208825376, 8296326612, 388694773720, 21155834296476, 1321107368127408, 93662776272057356, 7471576015922028248, 665418775120254506940, 65714704859545872003008, 7153378915302503698953860
Offset: 0

Views

Author

Seiichi Manyama, Jun 19 2022

Keywords

Crossrefs

Cf. A048801, A355101.
Cf. A355083.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=2*i*sum(j=0, i-1, stirling(i-1, j, 2)*v[j+1])); v;

Formula

a(0) = 1; a(n) = 2 * n * Sum_{k=0..n-1} Stirling2(n-1,k) * a(k).
a(n) = 2 * n * A355083(n-1) for n>0.

A355101 E.g.f. A(x) satisfies A(x) = 1 + 3 * x * A(exp(x) - 1).

Original entry on oeis.org

1, 3, 18, 189, 2952, 63225, 1759374, 61261200, 2595618720, 130963993263, 7734817065600, 527276606418837, 41005535326851456, 3602215645092352314, 354438336568129922052, 38776184401330464272910, 4686507224871009709115232, 622194587177907979874119473
Offset: 0

Views

Author

Seiichi Manyama, Jun 19 2022

Keywords

Crossrefs

Cf. A048801, A355100.
Cf. A355092.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=3*i*sum(j=0, i-1, stirling(i-1, j, 2)*v[j+1])); v;

Formula

a(0) = 1; a(n) = 3 * n * Sum_{k=0..n-1} Stirling2(n-1,k) * a(k).
a(n) = 3 * n * A355092(n-1) for n>0.

A355128 E.g.f. A(x) satisfies A(x) = 1 + x * A(2 * (exp(x) - 1)).

Original entry on oeis.org

1, 1, 4, 54, 1928, 167770, 34128972, 15867798142, 16621680303888, 38813463431274402, 200266228576991017940, 2265670919773168963168454, 55816752493202168837392763544, 2976116188645489878229876218205674, 341574630434025162744892242114482410332
Offset: 0

Views

Author

Seiichi Manyama, Jun 20 2022

Keywords

Crossrefs

Cf. A048801, A355100, A355131.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*sum(j=0, i-1, 2^j*stirling(i-1, j, 2)*v[j+1])); v;

Formula

a(0) = 1; a(n) = n * Sum_{k=0..n-1} 2^k * Stirling2(n-1,k) * a(k).
a(n) = n * A355131(n-1) for n>0.
Showing 1-8 of 8 results.

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

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