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 1-10 of 55 results. Next

A088715 G.f. satisfies: A(x*g(x)) = g(x) where g(x) is the g.f. of A088716.

Original entry on oeis.org

1, 1, 2, 7, 36, 240, 1926, 17815, 184916, 2116498, 26391700, 355405934, 5134778584, 79178537346, 1297633495518, 22522717498167, 412754532495252, 7965288555078018, 161475849044919996, 3431346397643014818
Offset: 0

Views

Author

Paul D. Hanna, Oct 12 2003

Keywords

Links

Crossrefs

Cf. A088716, A238223.

Programs

  • PARI
    a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(sum(k=1,n,(1-x*deriv(log(A)))^(-k)*x^k/k)));polcoeff(A,n) \\ Paul D. Hanna, Aug 31 2009
  • PARI
    a(n)=local(A=1+x); for(i=1, n, A=1+x*A^2/(A-x*deriv(A)+x*O(x^n))); polcoeff(A, n)
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Mar 20 2013

Formula

G.f.: Coefficient of x^n in A(x)^(n+1)/(n+1) = coefficient of x^n in A(x)^(n+2) = A088716(n).
G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/(1 - x*[A'(x)/A(x)])^n/n ). - Paul D. Hanna, Aug 31 2009
G.f. satisfies: A(x) = 1 + x*A(x)^2/(A(x) - x*A'(x)). - Paul D. Hanna, Mar 20 2013
a(n) ~ c * n! * n^2, where c = A238223 / exp(1) = 0.08017961462469262235245081077906956577... - Vaclav Kotesovec, Feb 21 2014

A238223 Decimal expansion of a constant related to A088716.

Original entry on oeis.org

2, 1, 7, 9, 5, 0, 7, 8, 9, 4, 4, 7, 1, 5, 1, 0, 6, 5, 4, 9, 2, 8, 2, 2, 8, 2, 2, 4, 4, 2, 3, 1, 9, 8, 2, 0, 8, 8, 6, 6, 0, 4, 5, 3, 9, 5, 6, 2, 9, 3, 9, 9, 6, 3, 4, 8, 1, 2, 3, 4, 0, 1, 7, 6, 2, 6, 5, 8, 7, 3, 3, 6, 2, 9, 2, 5, 3, 7, 0, 9, 4, 4, 9, 1, 2, 5, 9, 6, 3, 2, 2, 9, 8, 6, 2, 2, 9, 4, 5, 1, 4, 4, 8, 8, 9, 0
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 21 2014

Keywords

Examples

			0.21795078944715106549282282244231982088...

Links

Crossrefs

Cf. A088716, A088715, A158884, A159606, A182962, A193332, A156326, A338377.

Formula

Equals lim n->infinity A088716(n)/(n!*n^2).

A112916 Self-convolution of A088716, where a(n) = 2*A088716(n+1)/(n+2) for n>=0.

Original entry on oeis.org

1, 2, 7, 34, 207, 1496, 12420, 115938, 1198831, 13582010, 167187547, 2221174504, 31675372612, 482628099144, 7825665501852, 134562607924194, 2446051941152127, 46873289933397206, 944492559814284397
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2005

Keywords

Crossrefs

Cf. A088716, A112915, A112911.

Programs

  • PARI
    {a(n)=local(A=Mat(1),B);for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,B[i,j]=-j*(A^-1)[i-j,1]);));A=B); return(Vec(Ser(vector(n+1,i,(A^-1)[i,1]))^2)[n+1])}

A182962 E.g.f. satisfies: A(x) = exp( x/(1 - x*A'(x)/A(x)) ).

Original entry on oeis.org

1, 1, 3, 25, 433, 12501, 529531, 30495613, 2272643745, 211761416233, 24055076979091, 3267213865097601, 522451410607362193, 97120159467079471165, 20765771676360919883403, 5060640084128464622069221
Offset: 0

Views

Author

Paul D. Hanna, Jan 01 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 433*x^4/4! +...
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 621*x^6 + 5236*x^7 + 49680*x^8 +...+ A088716(n)*x^(n+1) +...
...
The coefficients of [x^n/n!] in the powers of e.g.f. A(x) begin:
A^1: [(1),(1), 3, 25, 433, 12501, 529531, 30495613, ...];
A^2: [1,(2),(8), 68, 1120, 30832, 1260544, 70737536, ...];
A^3: [1, 3,(15),(135), 2169, 57303, 2261439, 123523515, ...];
A^4: [1, 4, 24,(232),(3712), 94944, 3622336, 192461056, ...];
A^5: [1, 5, 35, 365, (5905),(147625), 5460475, 282185825, ...];
A^6: [1, 6, 48, 540, 8928, (220176),(7926336), 398625408, ...];
A^7: [1, 7, 63, 763, 12985, 318507,(11210479),(549313471), ...];
A^8: [1, 8, 80, 1040, 18304, 449728, 15551104,(743759360), ...];
...
In the above table, the coefficients in parenthesis are related by:
1*1 = 1; 8 = 2^2*2; 135 = 3^2*15; 3712 = 4^2*232; 147625 = 5^2*5905;
this illustrates: [x^n/n!] A(x)^n = n^2*[x^(n-1)/(n-1)!] A(x)^n.
...
Also note that the main diagonal in the above table begins:
[1*1, 2*1, 3*5, 4*58, 5*1181, 6*36696, 7*1601497, 8*92969920, ...];
this illustrates: [x^n/n!] A(x)^(n+1) = (n+1)*A156326(n).
...
Let G(x) denote the e.g.f. of A156326:
G(x) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1181*x^4/4! + 36696*x^5/5! +...
then G(x) satisfies: G(x) = A(x*G(x)) and A(x) = G(x/A(x)) where
G(x) = exp( Sum_{n>=1} n^2 * A156326(n-1)*x^n/n! ).
...

Links

Crossrefs

Cf. A088716, A300735, A300986, A300988, A300990, A300992.
Cf. A156326, A238223.

Programs

  • Mathematica
    m = 16; A[_] = 1;
Do[A[x_] = Exp[x/(1 - x A'[x]/A[x])] + O[x]^m, {m}];
CoefficientList[A[x], x] Range[0, m-1]! (* Jean-François Alcover, Oct 29 2019 *)
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(x/(1 - x*deriv(A)/A+x*O(x^n))));n!*polcoeff(A,n)}
  • PARI
    {a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0);A[#A]=((#A-1)*Vec(Ser(A)^(#A-1))[#A-1]-Vec(Ser(A)^(#A-1))[#A])/(#A-1));n!*A[n+1]}

Formula

E.g.f.: A(x) = exp(x*F(x)) where F(x) = 1 + x*F(x)*d/dx[x*F(x)] is the o.g.f. of A088716.
E.g.f. satisfies: [x^n/n!] A(x)^n = n^2*[x^(n-1)/(n-1)!] A(x)^n for n>=1.
E.g.f. satisfies: [x^n/n!] A(x)^(n+1) = (n+1)*A156326(n) for n>=0.
E.g.f.: A(x) = x/Series_Reversion(x*G(x)) where A(x*G(x)) = G(x) is the e.g.f. of A156326, which satisfies:
. G(x) = exp( Sum_{n>=1} n^2 * A156326(n-1)*x^n/n! ).
a(n) ~ c * (n!)^2 * n, where c = 0.21795078944715106549... (see A238223). - Vaclav Kotesovec, Feb 22 2014

A321087 O.g.f. A(x) satisfies: [x^n] exp(n*A(x)) * (1 - n*x/(1-x)) = 0, for n > 0.

Original entry on oeis.org

1, 2, 7, 37, 256, 2128, 20294, 216213, 2530522, 32165101, 440388103, 6454695553, 100786308221, 1669953080587, 29265149535076, 540884779563305, 10516595791609376, 214625521232021413, 4588068733776013386, 102541337542692407011, 2391813703854249362395, 58130860852912365134992, 1469860403455095402834628, 38611523432412179047238389
Offset: 1

Views

Author

Paul D. Hanna, Oct 27 2018

Keywords

Comments

It is remarkable that this sequence should consist entirely of integers.
Compare to: [x^n] exp(n*G(x)) * (1 - n*x) = 0, for n > 0, when G(x) = x + x*G(x)*G'(x), where G(x)/x is the o.g.f. of A088716.

Examples

			O.g.f.: A(x) = x + 2*x^2 + 7*x^3 + 37*x^4 + 256*x^5 + 2128*x^6 + 20294*x^7 + 216213*x^8 + 2530522*x^9 + 32165101*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n*A(x)) * (1 - n*x/(1-x)) begins:
n=1: [1, 0, 1, 28, 801, 30256, 1544425, 103604796, 8828789473, ...];
n=2: [1, 0, 0, 32, 1296, 55632, 2987200, 204441120, 17560833024, ...];
n=3: [1, 0, -3, 0, 1161, 67608, 4053645, 290790216, 25525161585, ...];
n=4: [1, 0, -8, -80, 0, 54304, 4333120, 344829888, 31719439360, ...];
n=5: [1, 0, -15, -220, -2655, 0, 3244825, 340694100, 34696521825, ...];
n=6: [1, 0, -24, -432, -7344, -115344, 0, 242169696, 32423666688, ...];
n=7: [1, 0, -35, -728, -14679, -316568, -6439475, 0, 22110305329, ...];
n=8: [1, 0, -48, -1120, -25344, -633792, -17406080, -451234944, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
(a) Differential Equation.
O.g.f. A(x) satisfies: A(x) = x/(1-x) + x*A(x)*A'(x) where
A'(x) = 1 + 4*x + 21*x^2 + 148*x^3 + 1280*x^4 + 12768*x^5 + 142058*x^6 + ...
A(x)*A'(x) = x + 6*x^2 + 36*x^3 + 255*x^4 + 2127*x^5 + 20293*x^6 + 216212*x^7 + 2530521*x^8 + 32165100*x^9 + ...
so that A(x) - x*A(x)*A'(x) = x/(1-x).
(b) Exponentiation.
exp(A(x)) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1129*x^4/4! + 37541*x^5/5! + 1813381*x^6/6! + 118181155*x^7/7! + 9890849585*x^8/8! + ...
exp(-A(x)) = 1 - x - 3*x^2/2! - 31*x^3/3! - 695*x^4/4! - 25221*x^5/5! - 1299779*x^6/6! - 88812907*x^7/7! - 7702826319*x^8/8! + ...

Crossrefs

Cf. A321086, A088716.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = -Vec( exp(m*x*Ser(A))*(1-m*x/(1-x +x^2*O(x^m))))[m+1]/m ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

O.g.f. A(x) satisfies: A(x) = x/(1-x) + x*A(x)*A'(x).

A385830 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k^2) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 20, 241, 4623, 130300, 5100750, 265780029, 17827454651, 1498498011875, 154408489507578, 19151761451917580, 2815820822235814540, 484383420815495253624, 96401320782466194458886, 21981036279413999807199045, 5693391431445001330242504699, 1662538953499888924638316487305
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Links

Crossrefs

Cf. A088716, A385831, A385832, A385833, A385834.
Cf. A385762, A385835.

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, (1+j^2)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) - x^2 * (d/dx A(x)) - x^3 * (d^2/dx^2 A(x)) ).

A300736 O.g.f. A(x) satisfies: A(x) = x*(1 - x*A'(x)) / (1 - 2*x*A'(x)).

Original entry on oeis.org

1, 1, 4, 24, 184, 1672, 17296, 198800, 2499200, 33992000, 496281344, 7731823616, 127946465280, 2240485196800, 41387447564800, 804353715776000, 16408115358117888, 350584123058300928, 7831051680901885952, 182550106828365115392, 4433782438058087202816, 112031844502468602085376, 2940834866411162315849728
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2018

Keywords

Comments

O.g.f. equals the logarithm of the e.g.f. of A300735.
The e.g.f. G(x) of A300735 satisfies: [x^n] G(x)^(2*n) = (n+1) * [x^(n-1)] G(x)^(2*n) for n>=1.

Examples

			O.g.f.: A(x) = x + x^2 + 4*x^3 + 24*x^4 + 184*x^5 + 1672*x^6 + 17296*x^7 + 198800*x^8 + 2499200*x^9 + 33992000*x^10 + 496281344*x^11 + 7731823616*x^12 + ...
where
A(x) = x*(1 - x*A'(x)) / (1 - 2*x*A'(x)).
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2/2! + 31*x^3/3! + 697*x^4/4! + 25761*x^5/5! + 1371691*x^6/6! + 97677343*x^7/7! + 8869533681*x^8/8! + 993709302337*x^9/9! + 134086553693011*x^10/10! + ... + A300735(n)*x^n/n! + ...
A'(x) = 1 + 2*x + 12*x^2 + 96*x^3 + 920*x^4 + 10032*x^5 + 121072*x^6 + 1590400*x^7 + 22492800*x^8 + 339920000*x^9 + 5459094784*x^10 + ...

Links

Crossrefs

Cf. A088716, A300987, A300989, A300991, A300993.
Cf. A300735, A088716, A300591, A296171, A300593, A300595.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1,n, A = x*(1-x*A')/(1-2*x*A' +x*O(x^n))); polcoeff(A,n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    /* [x^n] exp( 2*n * A(x) ) = (n + 1) * [x^(n-1)] exp( 2*n * A(x) ) */
{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(2*(#A-1))); A[#A] = ((#A)*V[#A-1] - V[#A])/(2*(#A-1)) ); polcoeff( log(Ser(A)), n)}
for(n=1, 25, print1(a(n), ", "))

Formula

O.g.f. A(x) satisfies: [x^n] exp( 2*n * A(x) ) = (n + 1) * [x^(n-1)] exp( 2*n * A(x) ) for n>=1.
a(n) ~ c * n! * n^3, where c = 0.0087891365985... - Vaclav Kotesovec, Mar 20 2018

A300987 O.g.f. A(x) satisfies: A(x) = x*(1 - 2*x*A'(x)) / (1 - 3*x*A'(x)).

Original entry on oeis.org

1, 1, 5, 36, 327, 3489, 42048, 559008, 8073243, 125328411, 2075525505, 36460943208, 676484058564, 13210384019292, 270753854165604, 5810388957096552, 130292809125319539, 3047472204302259711, 74227110587569392471, 1879966895740420683492, 49443968787368161215087, 1348661750106914651234385, 38107004920979745293594856, 1114125483618428275543280400, 33669232396216806674333898900
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2018

Keywords

Comments

O.g.f. equals the logarithm of the e.g.f. of A300986.
The e.g.f. G(x) of A300986 satisfies: [x^n] G(x)^(3*n) = (n+2) * [x^(n-1)] G(x)^(3*n) for n>=1.

Examples

			O.g.f.: A(x) = x + x^2 + 5*x^3 + 36*x^4 + 327*x^5 + 3489*x^6 + 42048*x^7 + 559008*x^8 + 8073243*x^9 + 125328411*x^10 + 2075525505*x^11 + ...
where
A(x) = x*(1 - 2*x*A'(x)) / (1 - 3*x*A'(x)).
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2/2! + 37*x^3/3! + 1009*x^4/4! + 44541*x^5/5! + 2799931*x^6/6! + 233188033*x^7/7! + 24562692897*x^8/8! + 3168510747769*x^9/9! + 488856473079571*x^10/10! + ... + A300986(n)*x^n/n! + ...
A'(x) = 1 + 2*x + 15*x^2 + 144*x^3 + 1635*x^4 + 20934*x^5 + 294336*x^6 + 4472064*x^7 + 72659187*x^8 + 1253284110*x^9 + 22830780555*x^10 + ...

Links

Crossrefs

Cf. A300986, A088716, A300736, A300989, A300991, A300993.
Cf. A296171, A300593, A300595.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = x*(1-2*x*A')/(1-3*x*A' +x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    /* [x^n] exp( 3*n * A(x) ) = (n + 2) * [x^(n-1)] exp( 3*n * A(x) ) */
{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(3*(#A-1))); A[#A] = ((#A+1)*V[#A-1] - V[#A])/(3*(#A-1)) ); polcoeff( log(Ser(A)), n)}
for(n=1, 25, print1(a(n), ", "))

Formula

O.g.f. A(x) satisfies: [x^n] exp( 3*n * A(x) ) = (n + 2) * [x^(n-1)] exp( 3*n * A(x) ) for n>=1.
a(n) ~ c * n! * n^5, where c = 0.00014640560804... - Vaclav Kotesovec, Mar 20 2018

A385762 G.f. A(x) satisfies A(x) = 1/(1 - x*A(x) - x^3*A''(x)).

Original entry on oeis.org

1, 1, 2, 9, 80, 1204, 27788, 918831, 41389972, 2443323132, 183303840972, 17050267807478, 1926895029660880, 260150110806399232, 41365993162914888760, 7652990621445212758255, 1630131235132495370561820, 396129991240222795968202788, 108937459572870420021782788268
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A000108, A088716, A385763, A385764, A385765.

Programs

  • Mathematica
    terms = 19; A[] = 0; Do[A[x] = 1/(1-x*A[x]-x^3*A''[x]) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 09 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, (1+sum(k=1, 2, stirling(2, k, 1)*j^k))*v[j+1]*v[i-j])); v;

Formula

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

A112915 Recurrence: a(n) = Sum_{k=0..n-1} (k+1)*(n-k)*a(k)*a(n-k-1) for n>0, with a(0)=1.

Original entry on oeis.org

1, 1, 4, 28, 272, 3312, 47872, 794880, 14840064, 306900736, 6953989120, 171200048128, 4548965384192, 129742326218752, 3953689388187648, 128215703582343168, 4409347536459988992, 160304460015345795072
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2005

Keywords

Links

Crossrefs

Cf. A088716, A112916, A238223.

Programs

  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,(k+1)*(n-k)*a(k)*a(n-k-1)))
  • PARI
    {a(n)=local(F=1+x+x*O(x^n));for(i=1,n,F=1+x*deriv(x*F)^2); return(polcoeff(F,n,x))}

Formula

A(x) = 1 + x*G(2*x)^2, where G(x) = g.f. of A088716, such that a(n) = 2^n*A088716(n)/(n+1) for n>=0.
a(n) = 2^(n-1)*A112916(n-1) for n>0.
G.f. satisfies: A(x) = 1 + x*(d/dx[x*A(x)])^2 = 1 + x*(A(x) + x*A'(x))^2.
a(n) ~ c * n * 2^n * n!, where c = A238223 = 0.21795078944715106549... - Vaclav Kotesovec, Aug 24 2017
Showing 1-10 of 55 results. Next

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

Content is available under The OEIS End-User License Agreement.