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 1-9 of 9 results.

A125282 G.f. satisfies: A(x) = Sum{n>=0} x^n * A(n*x).

Original entry on oeis.org

1, 1, 2, 5, 17, 80, 525, 4839, 62936, 1158785, 30277579, 1124649526, 59465788597, 4480380804517, 481401971074410, 73812092299235769, 16158739669470307453, 5052972095683109687920, 2257981256268589345121153
Offset: 0

Views

Author

Paul D. Hanna, Nov 29 2006

Keywords

Examples

			A(x) = 1 + x + 2*x^2 + 5*x^3 + 17*x^4 + 80*x^5 + 525*x^6 + 4839*x^7 +...
G.f. A(x) satisfies:
A(x) = 1 + x*A(x) + x^2*A(2x) + x^3*A(3x) + x^4*A(4x) + x^5*A(5x) +...
which leads to the recurrence illustrated by:
a(4) = 4^0*(1) + 3^1*(1) + 2^2*(2) + 1^3*(5) = 17;
a(5) = 5^0*(1) + 4^1*(1) + 3^2*(2) + 2^3*(5) + 1^4*(17) = 80;
a(6) = 6^0*(1) + 5^1*(1) + 4^2*(2) + 3^3*(5) + 2^4*(17) + 1^5*(80) = 525.

Links

Crossrefs

Cf. A125281 (variant), A210525.

Programs

  • Mathematica
    nmax = 20; aa = ConstantArray[0, nmax]; aa[[1]] = 1;Do[aa[[n]] = 1 + Sum[(n-k)^k*aa[[k]], {k, 1, n - 1}], {n, 2, nmax}]; Flatten[{1, aa}] (* Vaclav Kotesovec, Feb 23 2014 *)
  • PARI
    {a(n)=if(n==0,1,sum(k=0,n-1,(n-k)^k*a(k)))}

Formula

a(n) = Sum_{k=0..n-1} (n-k)^k * a(k) for n>0 with a(0)=1.
Limit n->infinity (a(n))^(1/n^2) = 3^(1/6). - Vaclav Kotesovec, Feb 24 2014
a(n) ~ c * 3^(n^2/6 - n/2), where c = 372374.41350200494715367264093778... if n=3k, c = 372374.41350258936507380006951913... if n=3k+1, and c = 372374.41350254286383864609841301... if n=3k+2. - Vaclav Kotesovec, Feb 24 2014

A218683 E.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n/n! * A(n*x).

Original entry on oeis.org

1, 1, 6, 69, 1432, 52065, 3202176, 324172597, 53099867136, 13888279032129, 5736880791449920, 3710252136325373661, 3729910949734728414624, 5792791811385586637686849, 13826260704559808415109532256, 50488064853691920270244556417445
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2012

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 69*x^3/3! + 1432*x^4/4! + 52065*x^5/5! +...
where
A(x) = 1 + x*A(x) + 2^2*x^2*A(2*x)/2! + 3^3*x^3*A(3*x)/3! + 4^4*x^4*A(4*x)/4! +...
which leads to the recurrence illustrated by:
a(1) = 1*1^1*(1) = 1;
a(2) = 1*2^2*(1) + 2*1^2*(1) = 6;
a(3) = 1*3^3*(1) + 3*2^3*(1) + 3*1^3*(6) = 69;
a(4) = 1*4^4*(1) + 4*3^4*(1) + 6*2^4*(6) + 4*1^4*(69) = 1432;
a(5) = 1*5^5*(1) + 5*4^5*(1) + 10*3^5*(6) + 10*2^5*(69) + 5*1^5*(1432) = 52065.

Crossrefs

Cf. A125281, A218682.

Programs

  • Mathematica
    Clear[a]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n,k] * (n-k)^n * a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Jul 03 2025 *)
  • PARI
    {a(n)=local(A=1);for(i=1,n,A=sum(k=0,n,k^k*x^k/k!*subst(A,x,k*x)+x*O(x^n)));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=if(n==0, 1, sum(k=0, n-1, binomial(n, k)*(n-k)^n*a(k)))}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n-1} binomial(n,k) * (n-k)^n * a(k) for n>0 with a(0)=1.
From Vaclav Kotesovec, Jul 03 2025: (Start)
a(n) ~ c * n! * 3^(n*(n+1)/6) / 2^(n/3), where
c = 26660.4828462502037986697479653... if mod(n,3) = 0,
c = 26660.4828502439141044598280378... if mod(n,3) = 1,
c = 26660.4828514026721542320888687... if mod(n,3) = 2. (End)

A230323 E.g.f.: Sum_{n>=0} x^n/n! * LambertW(-n*x)/(-n*x).

Original entry on oeis.org

1, 1, 3, 16, 149, 2196, 47167, 1380394, 52206345, 2457554248, 140408870651, 9563233300974, 765251809488157, 71012512562009500, 7552175961721086711, 911014865916673379026, 123562591612443767093393, 18704896422725902820936976, 3140424504257773679216307955
Offset: 0

Views

Author

Paul D. Hanna, Oct 16 2013

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 149*x^4/4! + 2196*x^5/5! +...
Let W(x) = LambertW(-x)/(-x), then
W(n*x) = Sum_{k>=0} n^k*(k+1)^(k-1)*x^k/k! and
W(n*x) = [ Sum_{k>=0} (n*k+1)^(k-1)*x^k/k! ]^n
where
A(x) = 1 + x*W(x) + x^2*W(2*x)/2! + x^3*W(3*x)/3! + x^4*W(4*x)/4! + x^5*W(5*x)/5! + x^6*W(6*x)/6! +...
Related expansions:
W(1*x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! +...
W(2*x) = 1 + 2*x + 12*x^2/2! + 128*x^3/3! + 2000*x^4/4! + 41472*x^5/5! +...
W(3*x) = 1 + 3*x + 27*x^2/2! + 432*x^3/3! + 10125*x^4/4! + 314928*x^5/5! +...
W(4*x) = 1 + 4*x + 48*x^2/2! + 1024*x^3/3! + 32000*x^4/4! + 1327104*x^5/5! +...
W(5*x) = 1 + 5*x + 75*x^2/2! + 2000*x^3/3! + 78125*x^4/4! + 4050000*x^5/5! +...
...
W(1*x) = (1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! +...)^1
W(2*x) = (1 + x + 5*x^2/2! + 7^2*x^3/3! + 9^3*x^4/4! + 11^4*x^5/5! +...)^2
W(3*x) = (1 + x + 7*x^2/2! + 10^2*x^3/3! + 13^3*x^4/4! + 16^4*x^5/5! +...)^3
W(4*x) = (1 + x + 9*x^2/2! + 13^2*x^3/3! + 17^3*x^4/4! + 21^4*x^5/5! +...)^4
W(5*x) = (1 + x + 11*x^2/2! + 16^2*x^3/3! + 21^3*x^4/4! + 26^4*x^5/5! +...)^5
...

Links

Crossrefs

Cf. A230321, A125281.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[n,k] * (n-k)^k * (k+1)^(k-1),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jul 29 2014 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)*(n-k)^k*(k+1)^(k-1))}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1);
A=sum(m=0,n,x^m/m!*sum(j=0,n,m^j*(j+1)^(j-1)*x^j/j! +x*O(x^n)) );
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1);
A=sum(m=0,n,x^m/m!*sum(j=0,n,(m*j+1)^(j-1)*x^j/j! +x*O(x^n))^m );
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=local(W=sum(m=0,n,(m+1)^(m-1)*x^m/m!)+x*O(x^n),A=1);
A=sum(m=0,n,x^m/m!*subst(W,x,m*x));
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=local(LambertW=serreverse(x*exp(x+x*O(x^n))),A=1);
A=1+sum(m=1,n,x^m/m!*subst(LambertW,x,-m*x)/(-m*x));
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * (n-k)^k * (k+1)^(k-1).
E.g.f.: Sum_{n>=0} x^n/n! * Sum_{k>=0} n^k*(k+1)^(k-1)*x^k/k!.
E.g.f.: Sum_{n>=0} x^n/n! * [ Sum_{k>=0} (n*k+1)^(k-1)*x^k/k! ]^n.

A385546 E.g.f. A(x) satisfies A(x) = Sum_{k>=0} x^k/k! * A(2*k*x).

Original entry on oeis.org

1, 1, 5, 73, 2841, 275841, 64191133, 34819498953, 43138054887665, 120180168811524673, 743714973197251893621, 10121932780068673030506249, 300492979488426431579059718281, 19324971141736799892467667226925889, 2676450415125002650227336306537165370253, 794234262095484293816451881720629809901365001
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2025

Keywords

Crossrefs

Cf. A125281, A385550.
Cf. A195067.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n,k] * 2^k * (n-k)^k * a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Jul 03 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (2*(i-j))^j*binomial(i, j)*v[j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (2*(n-k))^k * binomial(n,k) * a(k).
a(n) ~ c * n! * (1 + sqrt(2))^n * 2^(n*(n-3)/2), where c = 0.890249679316305512447698257069341300111198385329042058845194612667048446402873... - Vaclav Kotesovec, Jul 03 2025

A385550 E.g.f. A(x) satisfies A(x) = Sum_{k>=0} x^k/k! * A(-k*x).

Original entry on oeis.org

1, 1, -1, -8, -3, 516, 235, -336258, -3909367, 1462013704, 28386802071, -43272301150110, -3999150112115411, 7064064858964289580, 1224799123758409303091, -6776368902478388538739274, -3903339581644484604889375215, 32981625415507385681195122476432
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2025

Keywords

Crossrefs

Cf. A125281, A385546.
Cf. A385551.

Programs

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

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (k-n)^k * binomial(n,k) * a(k).

A192036 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n*A(n*x)^n/n!.

Original entry on oeis.org

1, 1, 3, 22, 317, 7976, 329167, 21511036, 2187830521, 343670351392, 83118756921371, 30891910810811084, 17606061819337679173, 15347380239670729742272, 20404520526924833144453623, 41254672227383167503175726876, 126484184787351358506375259745393
Offset: 0

Views

Author

Paul D. Hanna, Jun 21 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 317*x^4/4! + 7976*x^5/5! +...
The e.g.f. satisfies:
A(x) = 1 + x*A(x) + x^2*A(2*x)^2/2! + x^3*A(3*x)^3/3! + x^4*A(4*x)^4/4! +...

Crossrefs

Cf. A125281, A155806, A218682.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*subst(A,x,m*x+x*O(x^(n)))^m/m!));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

A196561 E.g.f. satisfies: A(x) = 1 + Sum{n>=1} x^n * A(n*x)^(1/n) / n!.

Original entry on oeis.org

1, 1, 3, 13, 87, 881, 13983, 358401, 15066943, 1031686513, 113292940383, 19641727327289, 5313784519456383, 2223596389445643993, 1429685952797008548463, 1404762001155914382481537, 2099828629077697954958843007
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 87*x^4/4! + 881*x^5/5! +...
where
A(x) = 1 + x*A(x) + x^2*A(2*x)^(1/2)/2! + x^3*A(3*x)^(1/3)/3! + x^4*A(4*x)^(1/4)/4! +...
Related expansions begin:
A(2*x)^(1/2) = 1 + x + 5*x^2/2! + 37*x^3/3! + 473*x^4/4! + 9881*x^5/5! +...
A(3*x)^(1/3) = 1 + x + 7*x^2/2! + 73*x^3/3! + 1387*x^4/4! + 44341*x^5/5! +...
A(4*x)^(1/4) = 1 + x + 9*x^2/2! + 121*x^3/3! + 3057*x^4/4! + 131921*x^5/5! +...
A(5*x)^(1/5) = 1 + x + 11*x^2/2! + 181*x^3/3! + 5711*x^4/4! + 310601*x^5/5! +...
A(6*x)^(1/6) = 1 + x + 13*x^2/2! + 253*x^3/3! + 9577*x^4/4! + 628681*x^5/5! +...

Crossrefs

Cf. A125281.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m/m!*subst(A,x,m*x+x*O(x^n))^(1/m)));n!*polcoeff(A,n)}

A385547 E.g.f. A(x) satisfies A(x) = Sum_{k>=0} x^k/k! * A(k^2*x).

Original entry on oeis.org

1, 1, 3, 22, 413, 18656, 2030287, 513423436, 300561564025, 398653905380896, 1192260459720446171, 7941386767782184832204, 117226647607145106003271333, 3808187092459275036716509871776, 271053748414379190468548152694690551, 42093494971632722160142716694680694172956
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2025

Keywords

Crossrefs

Cf. A125281, A385543.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n,k] * (n-k)^(2*k) * a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Jul 03 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (i-j)^(2*j)*binomial(i, j)*v[j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (n-k)^(2*k) * binomial(n,k) * a(k).

A196562 E.g.f. satisfies: A(x) = 1 + Sum{n>=1} x^n * A(n^2*x)^(1/n) / n!.

Original entry on oeis.org

1, 1, 3, 16, 197, 5556, 402727, 68650114, 28060721817, 25837746071608, 54301560755743691, 252957269930383300734, 2621503487300921168212357, 59671929727060536957652131604, 2977392188346587645059521680164959
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 197*x^4/4! + 5556*x^5/5! +...
where
A(x) = 1 + x*A(x) + x^2*A(4*x)^(1/2)/2! + x^3*A(9*x)^(1/3)/3! + x^4*A(16*x)^(1/4)/4! +...
Related expansions begin:
A(4*x)^(1/2) = 1 + 2*x + 20*x^2/2! + 392*x^3/3! + 20880*x^4/4! +...
A(9*x)^(1/3) = 1 + 3*x + 63*x^2/2! + 2700*x^3/3! + 335421*x^4/4! +...
A(16*x)^(1/4) = 1 + 4*x + 144*x^2/2! + 10816*x^3/3! + 2437376*x^4/4! +...
A(25*x)^(1/5) = 1 + 5*x + 275*x^2/2! + 32000*x^3/3! + 11413125*x^4/4! +...
A(36*x)^(1/6) = 1 + 6*x + 468*x^2/2! + 77976*x^3/3! + 40405392*x^4/4! +...

Crossrefs

Cf. A196561, A125281.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m/m!*subst(A,x,m^2*x+x*O(x^n))^(1/m)));n!*polcoeff(A,n)}
Showing 1-9 of 9 results.

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