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.

Previous Showing 11-20 of 55 results. Next

A300989 O.g.f. A(x) satisfies: A(x) = x*(1 - 3*x*A'(x)) / (1 - 4*x*A'(x)).

Original entry on oeis.org

1, 1, 6, 50, 520, 6312, 86080, 1288704, 20862720, 361454720, 6652338176, 129341001216, 2645494627328, 56734280221696, 1272300911597568, 29769957834147840, 725430667245355008, 18379623419316338688, 483476314203202945024, 13187069277429966733312, 372512001057014648537088, 10886129458069912361631744, 328776894530826384975593472
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2018

Keywords

Comments

O.g.f. equals the logarithm of the e.g.f. of A300988.
The e.g.f. G(x) of A300988 satisfies: [x^n] G(x)^(4*n) = (n+3) * [x^(n-1)] G(x)^(4*n) for n>=1.

Examples

			O.g.f.: A(x) = x + x^2 + 6*x^3 + 50*x^4 + 520*x^5 + 6312*x^6 + 86080*x^7 + 1288704*x^8 + 20862720*x^9 + 361454720*x^10 + ...
where
A(x) = x * (1 - 3*x*A'(x)) / (1 - 4*x*A'(x)).
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2/2! + 43*x^3/3! + 1369*x^4/4! + 69561*x^5/5! + 4991371*x^6/6! + 471516403*x^7/7! + 56029153713*x^8/8! + 8112993527089*x^9/9! + ... + A300988(n)*x^n/n! + ...
A'(x) = 1 + 2*x + 18*x^2 + 200*x^3 + 2600*x^4 + 37872*x^5 + 602560*x^6 + 10309632*x^7 + 187764480*x^8 + 3614547200*x^9 + ...

Links

Crossrefs

Cf. A300988, A088716, A300736, A300987, A300991, A300993.
Cf. A300591, A296171, A300593, A300595.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = x*(1-3*x*A')/(1-4*x*A' +x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    /* [x^n] exp( 4*n * A(x) ) = (n + 3) * [x^(n-1)] exp( 4*n * A(x) ) */
{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(4*(#A-1))); A[#A] = ((#A+2)*V[#A-1] - V[#A])/(4*(#A-1)) ); polcoeff( log(Ser(A)), n)}
for(n=1, 25, print1(a(n), ", "))

Formula

O.g.f. A(x) satisfies: [x^n] exp( 4*n * A(x) ) = (n + 3) * [x^(n-1)] exp( 4*n * A(x) ) for n>=1.
a(n) ~ c * n! * n^7, where c = 0.00000132855349... - Vaclav Kotesovec, Mar 20 2018

A300991 O.g.f. A(x) satisfies: A(x) = x * (1 - 4*x*A'(x)) / (1 - 5*x*A'(x)).

Original entry on oeis.org

1, 1, 7, 66, 769, 10405, 157540, 2609120, 46569365, 886686635, 17878075475, 379658470550, 8456459003700, 196924945517500, 4781934837995500, 120830066634026000, 3171336593264680125, 86326557180541530375, 2433904105428495204125, 70991247725936793222750, 2139844431980825211095625, 66589703922949982943093125, 2137375830643389417662717500, 70701647390083655222041600000
Offset: 1

Views

Author

Paul D. Hanna, Mar 19 2018

Keywords

Comments

O.g.f. equals the logarithm of the e.g.f. of A300990.
The e.g.f. G(x) of A300990 satisfies: [x^n] G(x)^(5*n) = (n+4) * [x^(n-1)] G(x)^(5*n) for n>=1.

Examples

			O.g.f.: A(x) = x + x^2 + 7*x^3 + 66*x^4 + 769*x^5 + 10405*x^6 + 157540*x^7 + 2609120*x^8 + 46569365*x^9 + 886686635*x^10 + ...
where
A(x) = x * (1 - 4*x*A'(x)) / (1 - 5*x*A'(x)).
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2/2! + 49*x^3/3! + 1777*x^4/4! + 101541*x^5/5! + 8140411*x^6/6! + 855134533*x^7/7! + 112545136929*x^8/8! + 17984228218057*x^9/9! + ... + A300990(n)*x^n/n! + ...
A'(x) = 1 + 2*x + 21*x^2 + 264*x^3 + 3845*x^4 + 62430*x^5 + 1102780*x^6 + 20872960*x^7 + 419124285*x^8 + 8866866350*x^9 + ...

Links

Crossrefs

Cf. A300990, A088716, A300736, A300987, A300989, A300993.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = x*(1-4*x*A')/(1-5*x*A' +x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    /* [x^n] exp( 5*n * A(x) ) = (n + 4) * [x^(n-1)] exp( 5*n * A(x) ) */
{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(5*(#A-1))); A[#A] = ((#A+3)*V[#A-1] - V[#A])/(5*(#A-1)) ); polcoeff( log(Ser(A)), n)}
for(n=1, 25, print1(a(n), ", "))

Formula

O.g.f. A(x) satisfies: [x^n] exp( 5*n * A(x) ) = (n + 4) * [x^(n-1)] exp( 5*n * A(x) ) for n>=1.
a(n) ~ c * n! * n^9, where c = 0.00000000758078051425272... - Vaclav Kotesovec, Mar 20 2018

A300993 O.g.f. A(x) satisfies: A(x) = x * (1 - 5*x*A'(x)) / (1 - 6*x*A'(x)).

Original entry on oeis.org

1, 1, 8, 84, 1080, 16056, 266256, 4816080, 93638016, 1937252160, 42339628800, 972303685632, 23365476089856, 585706819083264, 15276194983411200, 413695882240574976, 11612673418376392704, 337392794531354462208, 10133165365696293507072, 314252173854006410465280, 10053170842576476899524608, 331455812860465669006442496
Offset: 1

Views

Author

Paul D. Hanna, Mar 19 2018

Keywords

Comments

O.g.f. equals the logarithm of the e.g.f. of A300992.
The e.g.f. G(x) of A300992 satisfies: [x^n] G(x)^(6*n) = (n+5) * [x^(n-1)] G(x)^(6*n) for n>=1.

Examples

			O.g.f.: A(x) = x + x^2 + 8*x^3 + 84*x^4 + 1080*x^5 + 16056*x^6 + 266256*x^7 + 4816080*x^8 + 93638016*x^9 + 1937252160*x^10 + ...
where
A(x) = x * (1 - 5*x*A'(x)) / (1 - 6*x*A'(x)).
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2/2! + 55*x^3/3! + 2233*x^4/4! + 141201*x^5/5! + 12458731*x^6/6! + 1435102663*x^7/7! + 206465053425*x^8/8! + 35963535971233*x^9/9! + ... + A300992(n)*x^n/n! + ...
A'(x) = 1 + 2*x + 24*x^2 + 336*x^3 + 5400*x^4 + 96336*x^5 + 1863792*x^6 + 38528640*x^7 + 842742144*x^8 + 19372521600*x^9 + ...

Links

Crossrefs

Cf. A300992, A088716, A300736, A300987, A300989, A300991, A300993.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = x*(1-5*x*A')/(1-6*x*A' +x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    /* [x^n] exp( 6*n * A(x) ) = (n + 5) * [x^(n-1)] exp( 6*n * A(x) ) */
{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(6*(#A-1))); A[#A] = ((#A+4)*V[#A-1] - V[#A])/(6*(#A-1)) ); polcoeff( log(Ser(A)), n)}
for(n=1, 25, print1(a(n), ", "))

Formula

O.g.f. A(x) satisfies: [x^n] exp( 6*n * A(x) ) = (n + 5) * [x^(n-1)] exp( 6*n * A(x) ) for n>=1.
a(n) ~ c * n! * n^11, where c = 0.00000000002970897246102814... - Vaclav Kotesovec, Mar 20 2018

A317352 G.f. satisfies: A(x) = 1 + x*A(x) * ( d/dx x*A(x)^2 ).

Original entry on oeis.org

1, 1, 5, 42, 471, 6435, 102232, 1837630, 36719439, 805716679, 19239923577, 496514053880, 13769677836500, 408449335836132, 12906850662570996, 432942515731367894, 15367227978734187567, 575544844737119275935, 22685977410186834271463, 938867118118688412116554
Offset: 0

Views

Author

Paul D. Hanna, Jul 26 2018

Keywords

Examples

			O.g.f.: A(x) = 1 + x + 5*x^2 + 42*x^3 + 471*x^4 + 6435*x^5 + 102232*x^6 + 1837630*x^7 + 36719439*x^8 + 805716679*x^9 + ...
where A(x) = 1 + x*A(x)^3 + 2*x^2*A(x)^2*A'(x).
RELATED TABLE.
The table of coefficients of x^k/k! in exp( n*x*A(x)^2 ) / A(x) begins:
n=1: [1, 0, -5, -158, -7779, -563924, -56177105, -7318104450, ...];
n=2: [1, 1, 0, -94, -5968, -473688, -49352768, -6601523360, ...];
n=3: [1, 2, 7, 0, -3435, -354282, -40709709, -5723430444, ...];
n=4: [1, 3, 16, 130, 0, -199016, -29893568, -4657391616, ...];
n=5: [1, 4, 27, 302, 4541, 0, -16486865, -3372747590, ...];
n=6: [1, 5, 40, 522, 10416, 251976, 0, -1833979680, ...];
n=7: [1, 6, 55, 796, 17877, 567562, 20138467, 0, ...];
n=8: [1, 7, 72, 1130, 27200, 958968, 44592256, 2176638976, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.

Links

Crossrefs

Cf. A088716, A208961, A317353, A317354, A352236.

Programs

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

Formula

O.g.f. A(x) satisfies:
(1) A(x) = 1 + x*A(x) * ( d/dx x*A(x)^2 ).
(2) [x^n] exp( n * x*A(x)^2 ) / A(x) = 0 for n>0.
(3.a) [x^n] exp(-n * x*A(x)^2) * (2 - 1/A(x)) = 0 for n >= 1.
(3.b) [x^n] exp(-n^2 * x*A(x)^2) * (n + 1 - n/A(x)) = 0 for n >= 1.
(3.c) [x^n] exp(-n^(p+1) * x*A(x)^2) * (n^p + 1 - n^p/A(x)) = 0 for n>=1 and for fixed integer p >= 0.
a(n) ~ c * 2^n * n! * n^(3/2), where c = 0.188286926603706833845600622... - Vaclav Kotesovec, Aug 05 2018

A385833 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k^5) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 104, 25585, 26276091, 82191698776, 639369308538270, 10747798328839679301, 352216100969784522738455, 20799065226839989441184616755, 2079968920938449464603267217930862, 334987314655287149221766445992266495796, 83356568448492338030736248231384628286761124
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A088716, A385830, A385831, A385832, A385834.
Cf. A385765, A385843.

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^5)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) - x*Sum_{k=1..5} Stirling2(5,k) * x^k * (d^k/dx^k A(x)) ).

A385834 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k^6) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 200, 146401, 600098283, 9378336443140, 437583801957155730, 51482609496251191260549, 13496011632930307406903060651, 7172374406405634119759727327588155, 7172395923569361382696722735713532276498, 12706358411963754476880803069979932030145242780
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A088716, A385830, A385831, A385832, A385833.

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^6)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) - x*Sum_{k=1..6} Stirling2(6,k) * x^k * (d^k/dx^k A(x)) ).

A182304 G.f. satisfies: A(x) = x + d/dx A(x)^3/3.

Original entry on oeis.org

1, 1, 4, 25, 200, 1890, 20248, 240069, 3102000, 43226590, 644778520, 10237504810, 172274940656, 3061868388100, 57310884482160, 1126953452418765, 23230411494361920, 501015558669312390, 11285603304699928600, 265077311648541268030, 6482435029801210815280
Offset: 1

Views

Author

Paul D. Hanna, Apr 23 2012

Keywords

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 200*x^5 + 1890*x^6 + 20248*x^7 +...
Related expansions.
A'(x) = 1 + 2*x + 12*x^2 + 100*x^3 + 1000*x^4 + 11340*x^5 + 141736*x^6 +...
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 58*x^5 + 466*x^6 + 4380*x^7 + 46501*x^8 +...
A(x)^3 = x^3 + 3*x^4 + 15*x^5 + 100*x^6 + 810*x^7 + 7593*x^8 + 80023*x^9 +...
d/dx A(x)^3 = 3*x^2 + 12*x^3 + 75*x^4 + 600*x^5 + 5670*x^6 + 60744*x^7 +...

Links

Crossrefs

Cf. A245118, A088716.

Programs

  • PARI
    {a(n)=local(A=x+x^2+x*O(x^n));for(i=1,n,A=x+deriv(A^3/3+O(x^31)));polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))
  • PARI
    /* From g.f. F(x) of A245118 (Paul D. Hanna, Jul 27 2014): */
{a(n)=local(F=1+x); for(i=1, n, F = 1 + x*F^3/(F - x*F' +x*O(x^n))); polcoeff(serreverse(x/F), n)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. satisfies: A(x) = x + A'(x)*A(x)^2.
a(n) ~ c * n! * n^3, where c = 0.0147556981601927885048672335828437... - Vaclav Kotesovec, Feb 22 2014
G.f.: A(x) = Series_Reversion(x/F(x)) where F(x) = 1 + x*F(x)^3/(F(x) - x*F'(x)) is the g.f. of A245118. - Paul D. Hanna, Jul 27 2014

A319938 O.g.f. A(x) satisfies: [x^n] exp(-n*A(x)) / (1 - n*x) = 0, for n > 0.

Original entry on oeis.org

1, 1, 3, 18, 165, 2019, 30688, 554784, 11591649, 274313325, 7242994143, 210931834662, 6713206636084, 231754182524900, 8624280230971980, 344124280164153056, 14656294893872323449, 663624782214112471329, 31833832291287920426617, 1612762327644980719082470, 86050799297228500838101677, 4823357354919905244973170883, 283375597845431500054861239512
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 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 + x^2 + 3*x^3 + 18*x^4 + 165*x^5 + 2019*x^6 + 30688*x^7 + 554784*x^8 + 11591649*x^9 + 274313325*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n*A(x)) / (1 - n*x) begins:
n=1: [1, 0, -1, -16, -423, -19616, -1444625, -154014624, ...];
n=2: [1, 0, 0, -20, -768, -38832, -2895680, -308705280, ...];
n=3: [1, 0, 3, 0, -783, -53568, -4309605, -465802704, ...];
n=4: [1, 0, 8, 56, 0, -50144, -5307200, -616050432, ...];
n=5: [1, 0, 15, 160, 2265, 0, -4729025, -711963600, ...];
n=6: [1, 0, 24, 324, 6912, 145584, 0, -613885824, ...];
n=7: [1, 0, 35, 560, 15057, 460768, 13696795, 0, ...];
n=8: [1, 0, 48, 880, 28032, 1050432, 44437120, 1769051136, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 529*x^4/4! + 22581*x^5/5! + 1598011*x^6/6! + 166508413*x^7/7! + 23765885025*x^8/8! + ...
exp(-A(x)) = 1 - x - x^2/2! - 13*x^3/3! - 359*x^4/4! - 17501*x^5/5! - 1326929*x^6/6! - 143902249*x^7/7! - 21072159247*x^8/8! + ...

Links

Crossrefs

Cf. A088716, A321085, A319939, A319940, A320417.

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 +x^2*O(x^m)))[m+1]/m ); A[n]}
for(n=1,30,print1(a(n),", "))

Formula

a(n) ~ c * n^(n-1), where c = 0.335949071234... - Vaclav Kotesovec, Oct 22 2020

A385763 G.f. A(x) satisfies A(x) = 1/(1 - x*A(x) - x^4*A'''(x)).

Original entry on oeis.org

1, 1, 2, 5, 44, 1188, 74880, 9211479, 1962123260, 665169218468, 337242780292376, 243827199998597254, 242120748323922920272, 320325994582940359050400, 550640627320172764415124000, 1204251372776149567847238889047, 3291219553094816112273747054673476
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A000108, A088716, A385762, A385764, A385765.

Programs

  • Mathematica
    terms = 17; A[] = 0; Do[A[x] = 1/(1-x*A[x]-x^4*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, 3, stirling(3, k, 1)*j^k))*v[j+1]*v[i-j])); v;

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + 2*k - 3*k^2 + k^3) * a(k) * a(n-1-k).

A385764 G.f. A(x) satisfies A(x) = 1/(1 - x*A(x) - x^5*A''''(x)).

Original entry on oeis.org

1, 1, 2, 5, 14, 378, 46500, 16879869, 14229776750, 23948731244678, 72492823741526156, 365581334105823084634, 2896500982661242290253612, 34419121542689992919239814260, 590735385934420874267059790772360, 14193599152271246770955912922939691797, 465024889616667096875210999651863472880846
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A000108, A088716, A385762, A385763, A385765.

Programs

  • Mathematica
    terms = 17; A[] = 0; Do[A[x] = 1/(1-x*A[x]-x^5*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, 4, stirling(4, k, 1)*j^k))*v[j+1]*v[i-j])); v;

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (1 - 6*k + 11*k^2 - 6*k^3 + k^4) * a(k) * a(n-1-k).
Previous Showing 11-20 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.