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 1-7 of 7 results.

A384822 G.f. A(x) satisfies 1/x^5 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+4).

Original entry on oeis.org

1, 1, 5, 19, 109, 598, 3592, 22110, 140467, 911136, 6014277, 40260501, 272682397, 1865181921, 12866239311, 89403333632, 625211046931, 4396844409898, 31075863324446, 220618909826500, 1572549447431889, 11249693613964519, 80743512234554655, 581272589032594530, 4196118995069449989
Offset: 0

Views

Author

Paul D. Hanna, Jun 10 2025

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 19*x^3 + 109*x^4 + 598*x^5 + 3592*x^6 + 22110*x^7 + 140467*x^8 + 911136*x^9 + 6014277*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1,1,0,0]); for(i=1, n, A = concat(A, 0);
    A[#A-2] = -polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+4) ), #A-9)); A[n+1]}
    for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1/x^5 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+4).
(2) x = Sum_{n=-oo..+oo, n<>0} (-1/A(x))^n * x^((n-2)*(n-3)) / (1 - x^n)^(n-4).
a(n) ~ c * d^n / n^(3/2), where d = 7.687452504111926947946743863677977... and c = 0.2779337748307189293421456249838... - Vaclav Kotesovec, Jun 11 2025

A384823 G.f. A(x) satisfies -1/x^11 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+6).

Original entry on oeis.org

1, 1, 4, 28, 173, 1262, 9593, 75928, 618342, 5149640, 43650123, 375347585, 3266282211, 28709930633, 254526671024, 2273271614848, 20435110855838, 184745786960642, 1678668998195885, 15321962225034079, 140418372363945954, 1291587696225346583, 11919771215919819476, 110338977972166474055
Offset: 0

Views

Author

Paul D. Hanna, Jun 10 2025

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 173*x^4 + 1262*x^5 + 9593*x^6 + 75928*x^7 + 618342*x^8 + 5149640*x^9 + 43650123*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1,1,0,0,0]); for(i=1, n, A = concat(A, 0);
    A[#A-3] = polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+6) ), #A-16)); A[n+1]}
    for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) -1/x^11 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+6).
(2) -x = Sum_{n=-oo..+oo, n<>0} (-1/A(x))^n * x^((n-3)*(n-4)) / (1 - x^n)^(n-6).
a(n) ~ c * d^n / n^(3/2), where d = 9.887717015668710733345454711929087306... and c = 0.160435430066288197856237263106693... - Vaclav Kotesovec, Jun 11 2025

A384824 G.f. A(x) satisfies 1/x^19 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+8).

Original entry on oeis.org

1, 1, 5, 38, 319, 2871, 27507, 273925, 2808973, 29457644, 314470771, 3405995019, 37334767867, 413397265017, 4617060957512, 51951448775027, 588371324004508, 6701761863368579, 76723673176823126, 882342098781937683, 10188542630975395255, 118082022786322630334, 1373108879790849494070
Offset: 0

Views

Author

Paul D. Hanna, Jun 10 2025

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 319*x^4 + 2871*x^5 + 27507*x^6 + 273925*x^7 + 2808973*x^8 + 29457644*x^9 + 314470771*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1,1,0,0,0,0]); for(i=1, n, A = concat(A, 0);
    A[#A-4] = -polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+8) ), #A-25)); A[n+1]}
    for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1/x^19 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+8).
(2) x = Sum_{n=-oo..+oo, n<>0} (-1/A(x))^n * x^((n-4)*(n-5)) / (1 - x^n)^(n-8).
a(n) ~ c * d^n / n^(3/2), where d = 12.46033620173328231233579215988893957838459959... and c = 0.113752375605091798753361983956448030623... - Vaclav Kotesovec, Jun 11 2025

A384825 G.f. A(x) satisfies -1/x^29 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+10).

Original entry on oeis.org

1, 1, 6, 54, 542, 5955, 69114, 835140, 10391843, 132262619, 1713785727, 22531557603, 299817809184, 4030217936308, 54646151953660, 746513545616000, 10264746883787021, 141955200254335604, 1973170863256461516, 27551902179444882489, 386288077655575999571, 5435910477286670671340
Offset: 0

Views

Author

Paul D. Hanna, Jun 10 2025

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 54*x^3 + 542*x^4 + 5955*x^5 + 69114*x^6 + 835140*x^7 + 10391843*x^8 + 132262619*x^9 + 1713785727*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1,1,0,0,0,0,0]); for(i=1, n, A = concat(A, 0);
    A[#A-5] = polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+10) ), #A-36)); A[n+1]}
    for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) -1/x^29 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+10).
(2) -x = Sum_{n=-oo..+oo, n<>0} (-1/A(x))^n * x^((n-5)*(n-6)) / (1 - x^n)^(n-10).
a(n) ~ c * d^n / n^(3/2), where d = 15.130878695250901787504105640277512076291321821... and c = 0.088532592960846902874974330489987793829057... - Vaclav Kotesovec, Jun 11 2025

A384826 G.f. A(x) satisfies 1/x^41 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+12).

Original entry on oeis.org

1, 1, 7, 73, 861, 11112, 151828, 2159179, 31627690, 473917665, 7230164079, 111926802631, 1753762735460, 27760507986844, 443257137593369, 7130838718144623, 115469073853104486, 1880570694656739472, 30784302913287253256, 506228988080918570208, 8358750672258509735440, 138528877561300962357350
Offset: 0

Views

Author

Paul D. Hanna, Jun 10 2025

Keywords

Examples

			G.f.: A(x) = 1 + x + 7*x^2 + 73*x^3 + 861*x^4 + 11112*x^5 + 151828*x^6 + 2159179*x^7 + 31627690*x^8 + 473917665*x^9 + 7230164079*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1,1,0,0,0,0,0,0]); for(i=1, n, A = concat(A, 0);
    A[#A-6] = -polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+12) ), #A-49)); A[n+1]}
    for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1/x^41 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+12).
(2) x = Sum_{n=-oo..+oo, n<>0} (-1/A(x))^n * x^((n-6)*(n-7)) / (1 - x^n)^(n-12).
a(n) ~ c * d^n / n^(3/2), where d = 17.821078213117779013059276484226766696509894506... and c = 0.072486824411461280676499747682168909434267... - Vaclav Kotesovec, Jun 11 2025

A384827 G.f. A(x) satisfies -1/x^55 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+14).

Original entry on oeis.org

1, 1, 8, 95, 1288, 19116, 300511, 4918268, 82918049, 1430142380, 25115651237, 447578072658, 8073426806649, 147122009148252, 2704441907759235, 50088849266618466, 933792151007378231, 17509062834076661230, 329985690688947517626, 6247533413700369107192, 118768564127167799819733
Offset: 0

Views

Author

Paul D. Hanna, Jun 10 2025

Keywords

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 95*x^3 + 1288*x^4 + 19116*x^5 + 300511*x^6 + 4918268*x^7 + 82918049*x^8 + 1430142380*x^9 + 25115651237*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1,1,0,0,0,0,0,0,0]); for(i=1, n, A = concat(A, 0);
    A[#A-7] = polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+14) ), #A-64)); A[n+1]}
    for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) -1/x^55 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+14).
(2) -x = Sum_{n=-oo..+oo, n<>0} (-1/A(x))^n * x^((n-7)*(n-8)) / (1 - x^n)^(n-14).
a(n) ~ c * d^n / n^(3/2), where d = 20.5190724870235230419993391970202418416256614273528... and c = 0.06135641554365612129872100433075021800206923942... - Vaclav Kotesovec, Jun 11 2025

A384828 G.f. A(x) satisfies 1/x^71 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+16).

Original entry on oeis.org

1, 1, 9, 120, 1839, 30862, 548783, 10160786, 193811734, 3782270289, 75158649892, 1515578476370, 30935212293083, 637920390487505, 13269865608471203, 278121828806207328, 5867506406619195047, 124502776024601555996, 2655381364988431518262, 56892952987400631546208, 1223972213493916563960331
Offset: 0

Views

Author

Paul D. Hanna, Jun 10 2025

Keywords

Examples

			G.f.: A(x) = 1 + x + 9*x^2 + 120*x^3 + 1839*x^4 + 30862*x^5 + 548783*x^6 + 10160786*x^7 + 193811734*x^8 + 3782270289*x^9 + 75158649892*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1,1,0,0,0,0,0,0,0,0]); for(i=1, n, A = concat(A, 0);
    A[#A-8] = -polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+16) ), #A-81)); A[n+1]}
    for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1/x^71 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+16).
(2) x = Sum_{n=-oo..+oo, n<>0} (-1/A(x))^n * x^((n-8)*(n-9)) / (1 - x^n)^(n-16).
a(n) ~ c * d^n / n^(3/2), where d = 23.2218466497883684132359544378917382382303363986... and c = 0.05318473987345007866210446949223464954972731... - Vaclav Kotesovec, Jun 11 2025
Showing 1-7 of 7 results.