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.

A125281 E.g.f. satisfies: A(x) = Sum{n>=0} x^n * A(n*x)/n!.

Original entry on oeis.org

1, 1, 3, 16, 149, 2316, 59047, 2429554, 159549945, 16557985432, 2693862309131, 682199144788734, 267277518618047797, 161130714885281760100, 148762112860064623199295, 209444428223095096806228346, 447998198975235291015396393713, 1450973400598977755884988875863216
Offset: 0

Views

Author

Paul D. Hanna, Nov 29 2006, Sep 22 2007

Keywords

Examples

			A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 149*x^4/4! + 2316*x^5/5! +...
where
A(x) = 1 + x*A(x) + x^2*A(2*x)/2! + x^3*A(3*x)/3! + x^4*A(4*x)/4! + x^5*A(5*x)/5! +...
which leads to the recurrence illustrated by:
a(3) = 1*3^0*(1) + 3*2^1*(1) + 3*1^2*(3) = 16;
a(4) = 1*4^0*(1) + 4*3^1*(1) + 6*2^2*(3) + 4*1^3*(16) = 149;
a(5) = 1*5^0*(1) + 5*4^1*(1) + 10*3^2*(3) + 10*2^3*(16) + 5*1^4*(149) = 2316.

Crossrefs

Cf. A230323, A125282, A218683.

Programs

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

Formula

a(n) = Sum_{k=0..n-1} C(n,k)*(n-k)^k * a(k) for n>0 with a(0)=1.

A195067 G.f. A(x) satisfies A(x) = Sum{n>=0} x^n * A(2*n*x).

Original entry on oeis.org

1, 1, 3, 17, 191, 4261, 189123, 16723689, 2949213319, 1037964817357, 729449200732395, 1024041038817726353, 2872628913886690237679, 16105674069113302453209781, 180504701103754829110217971731, 4044484405239396750189431682523833
Offset: 0

Views

Author

Paul D. Hanna, Sep 08 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 191*x^4 + 4261*x^5 +...
where:
A(x) = 1 + x*A(2*x) + x^2*A(4*x) + x^3*A(6*x) + x^4*A(8*x) + x^5*A(10*x) +...

Crossrefs

Cf. A125282, A385546.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[2^k * (n-k)^k * a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Jul 03 2025 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(k=1, n, A=1+sum(j=1, n, x^j*subst(A,x,2*j*x))); polcoeff(A, n)}
  • PARI
    {a(n)=if(n==0, 1, sum(k=0, n-1, 2^k*(n-k)^k*a(k)))}

Formula

a(n) = Sum_{k=0..n-1} 2^k*(n-k)^k * a(k) for n>0 with a(0)=1.
a(n) ~ c * (1 + sqrt(3))^n * 2^(n*(n-3)/2), where c = 0.9296543230172164460137009343716233391546324099495685771220234877636263909188... - Vaclav Kotesovec, Jul 03 2025

A210525 G.f.: A(x) = Sum_{n>=0} x^n * A(n*x) * A(-n*x).

Original entry on oeis.org

1, 1, 1, 2, 5, 17, 129, 823, 16549, 203762, 9057969, 229117047, 20251050469, 1099941609906, 182745344595201, 21826976698036317, 6625997170286326085, 1760792727174811634066, 963263479575497024110193, 572732462137043288595101258, 560867188756075075604736149189
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 17*x^5 + 129*x^6 + 823*x^7 +...
such that
A(x) = 1 + x*A(x)*A(-x) + x^2*A(2*x)*A(-2*x) + x^3*A(3*x)*A(-3*x) + x^4*A(4*x)*A(-4*x) + x^5*A(5*x)*A(-5*x) +...
where
A(x)*A(-x) = 1 + x^2 + 7*x^4 + 230*x^6 + 31667*x^8 + 17739221*x^10 + 40061321883*x^12 + 363330478847435*x^14 +...

Crossrefs

Cf. A125282, A196196.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=sum(k=0, n, x^k*subst(A,x,k*x +x*O(x^n))*subst(A,x,-k*x +x*O(x^n)))  );polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A385544 G.f. A(x) satisfies A(x) = 1 + Sum_{k>=1} k * x^k * A(k*x).

Original entry on oeis.org

1, 1, 3, 10, 47, 309, 2846, 37021, 681653, 17809832, 661559047, 34979604911, 2635492535328, 283176079413813, 43418520087991775, 9505069880265893234, 2972315533856872301859, 1328214006502208982129889, 848461208731369665554869614, 775048547251487987362405899361
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2025

Keywords

Crossrefs

Cf. A125282, A385548.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(n-k)^(k+1)*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)^(j+1)*v[j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (n-k)^(k+1) * a(k).
From Vaclav Kotesovec, Jul 03 2025: (Start)
a(n) ~ c * 3^(n*(n-1)/6), where
c = 151875.34106580399103962024875... if mod(n,3) = 0,
c = 151875.34107021815371382907842... if mod(n,3) = 1,
c = 151875.34106681673681179782608... if mod(n,3) = 2. (End)

A385543 G.f. A(x) satisfies A(x) = Sum_{k>=0} x^k * A(k^2*x).

Original entry on oeis.org

1, 1, 2, 7, 49, 676, 18861, 1062533, 121557594, 28281916427, 13399862563765, 12949857822909156, 25549330363139585961, 103025771800413460066681, 849971455496325163128172498, 14359775106466928789344919850719, 497276944869002836686738999984515113
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2025

Keywords

Crossrefs

Cf. A125282, A385547.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(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)*v[j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (n-k)^(2*k) * a(k).
From Vaclav Kotesovec, Jul 03 2025: (Start)
a(n) ~ c * n! * 3^(n*(n-4)/3) / 2^(n/3), where
c = 438919.4178887847632978930903514036169636302175176... if mod(n,3) = 0,
c = 438919.4215235929223401081041169940935227575106084... if mod(n,3) = 1,
c = 438919.4025215529290127441106624079221416448856280... if mod(n,3) = 2. (End)

A385551 G.f. A(x) satisfies A(x) = Sum_{k>=0} x^k * A(-k*x).

Original entry on oeis.org

1, 1, 0, -1, -1, 4, 3, -147, -770, 15721, 107475, -10571326, -227719699, 23388067045, 997369658304, -266390905549461, -23979310388230253, 11854227262950292612, 2477760824989752459951, -2291696206079126389222423, -973819883013544085453392378, 1953283201528648806346685956669
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2025

Keywords

Crossrefs

Cf. A125282, A195067.
Cf. A385550.

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*v[j+1])); v;

Formula

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

A209238 G.f.: A(x) = Sum_{n>=0} x^n * A(n*x) / A(x)^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 33, 261, 3125, 54281, 1360227, 49213679, 2551724168, 189734190172, 20202532257783, 3077389342754387, 670727373110373402, 209089605110085626174, 93228754516671937710239, 59465333534426105454885059, 54262238803939167296498684986
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 6*x^5 + 33*x^6 + 261*x^7 + 3125*x^8 +...
The table of coefficients of x^k in A(n*x)/A(x)^n begin:
  n=0: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
  n=1: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
  n=2: [1, 0, 1, 2, 18, 128, 1704, 28908, 729461, 26121996, ...];
  n=3: [1, 0, 3, 8, 102, 1032, 20052, 502104, 18852177, ...];
  n=4: [1, 0, 6, 20, 333, 4404, 113292, 3769344, 188468286, ...];
  n=5: [1, 0, 10, 40, 825, 13504, 433060, 17986560, 1123704270, ...];
  n=6: [1, 0, 15, 70, 1725, 33684, 1294525, 64474290, ...];
  n=7: [1, 0, 21, 112, 3213, 72912, 3266599, 189725472, ...];
  n=8: [1, 0, 28, 168, 5502, 142296, 7282212, 483225336, ...];
  n=9: [1, 0, 36, 240, 8838, 256608, 14768652, 1102246128,  ...]; ...
This sequence equals the antidiagonal sums of the above table.

Crossrefs

Cf. A125282.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=sum(k=0, n, x^k*subst(A,x,k*x +x*O(x^n))/A^(k))  );polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A227222 G.f. satisfies: A(x) = Sum{n>=0} A( n*x/(1-n*x) ) * x^n, with A(0)=1.

Original entry on oeis.org

1, 1, 2, 6, 27, 172, 1508, 18107, 297532, 6694132, 206841391, 8816277300, 520844677834, 42854370882379, 4933351077116176, 797908056582772334, 181972606629594221271, 58701383528452842764544, 26853636463946258949427440, 17457976736153040916394583563
Offset: 0

Views

Author

Paul D. Hanna, Sep 19 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 27*x^4 + 172*x^5 + 1508*x^6 +...
where
A(x) = 1 + A(x/(1-x))*x + A(2*x/(1-2*x))*x^2 + A(3*x/(1-3*x))*x^3 + A(4*x/(1-4*x))*x^4 + A(5*x/(1-5*x))*x^5 +...

Crossrefs

Cf. A125282.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,subst(A,x,m*x/(1-m*x+x*O(x^n)))*x^m));polcoeff(A,n)}
for(n=0,21,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} x^n * Sum{k=0..n} a(k)*(n-k)^k/(1 - (n-k)*x)^k = Sum_{n>=0} a(n)*x^n.

A385585 G.f. A(x) satisfies A(x) = Sum_{k>=0} (k*x)^k * A(k*x).

Original entry on oeis.org

1, 1, 5, 40, 457, 7101, 148270, 4206121, 165267951, 9117777074, 709325010385, 77906424970811, 12109278363587036, 2670187179684919761, 836451775445907622685, 372646977140600929476104, 236390047765997660237447061, 213719117789650238860723125601
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2025

Keywords

Crossrefs

Cf. A125282, A385544.
Cf. A218683.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(n-k)^n * a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Jul 04 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)^i*v[j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (n-k)^n * a(k).
From Vaclav Kotesovec, Jul 04 2025: (Start)
a(n) ~ c * 3^(n*(n+3)/6), where
c = 14331.87392277329... if mod(n,3) = 0,
c = 14331.87383811849... if mod(n,3) = 1,
c = 14331.87405112061... if mod(n,3) = 2. (End)
Showing 1-9 of 9 results.

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