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 10 results.

A303290 G.f. A(x) satisfies: 2 = Sum_{n>=0} (1/2^n) * (1+x)^(n^2) / A(x)^n.

Original entry on oeis.org

1, 3, 15, 225, 6003, 223029, 10403175, 577700889, 37009173207, 2679339499305, 216031850406327, 19187294118006057, 1861057604220294591, 195742656849628038465, 22192660352433291780159, 2698458809215198981964481, 350326879575505922875480047, 48370384900519379918253881361, 7078145146554395463373624118319, 1094300840117324691452685873392145
Offset: 0

Views

Author

Paul D. Hanna, Apr 20 2018

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 225*x^3 + 6003*x^4 + 223029*x^5 + 10403175*x^6 + 577700889*x^7 + 37009173207*x^8 + 2679339499305*x^9 + 216031850406327*x^10 + ...
such that A = A(x) satisfies:
2 = 1 + (1+x)/(2*A) + (1+x)^4/(2*A)^2 + (1+x)^9/(2*A)^3 + (1+x)^16/(2*A)^4 + (1+x)^25/(2*A)^5 + (1+x)^36/(2*A)^6 + (1+x)^49/(2*A)^7 + (1+x)^64/(2*A)^8 + ...

Links

Crossrefs

Cf. A303291, A303292, A303058.
Cf. A325286.

Programs

  • PARI
    /* Find A(x) that satisfies the continued fraction: */
{a(n) = my(A=[1],q=1+x,CF=1); for(i=1,n, A=concat(A,0); m=#A; for(k=0, m, CF = 1/(1 - q^(4*m-4*k+1)/(2*Ser(A) - q^(2*m-2*k+1)*(q^(2*m-2*k+2) - 1)*CF)) ); A[#A] = Vec(CF)[#A]/2 );A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: 2 = 1/(1 - q/(2*A(x) - q*(q^2-1)/(1 - q^5/(2*A(x) - q^3*(q^4-1)/(1 - q^9/(2*A(x) - q^5*(q^6-1)/(1 - q^13/(2*A(x) - q^7*(q^8-1)/(1 - ...))))))))), where q = (1+x), a continued fraction due to a partial elliptic theta function identity.
G.f.: 2 = Sum_{n>=0} (1+x)^n/(2^n*A(x)^n) * Product_{k=1..n} (2*A(x) - (1+x)^(4*k-3)) / (2*A(x) - (1+x)^(4*k-1)), due to a q-series identity.
a(n) ~ c * 2^(2*n) * n^n / (exp(n) * log(2)^(2*n)), where c = 0.339650521725496... - Vaclav Kotesovec, Oct 06 2020

A301929 G.f. A(x) satisfies: x = Sum_{n>=1} (1+x)^(n^2) * x^n / A(x)^n.

Original entry on oeis.org

1, 2, 3, 5, 12, 37, 138, 595, 2843, 14844, 83212, 496473, 3128584, 20707672, 143342216, 1034075244, 7752274237, 60251286521, 484483164365, 4023459643530, 34455215830001, 303839675537827, 2755675307738286, 25675275100067189, 245502965520844801, 2406797239543382867, 24170220195274548727, 248441483165679473094, 2611787614440970964621
Offset: 0

Views

Author

Paul D. Hanna, May 04 2018

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 3*x^2 + 5*x^3 + 12*x^4 + 37*x^5 + 138*x^6 + 595*x^7 + 2843*x^8 + 14844*x^9 + 83212*x^10 + 496473*x^11 + 3128584*x^12 + ...
such that
x = (1+x)*x/A(x) + (1+x)^4*x^2/A(x)^2 + (1+x)^9*x^3/A(x)^3 + (1+x)^16*x^4/A(x)^4 + (1+x)^25*x^5/A(x)^5 + (1+x)^36*x^6/A(x)^6 + (1+x)^49*x^7/A(x)^7 + ...

Links

Crossrefs

Cf. A301927, A303058, A107595.

Programs

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

Formula

G.f.: x = Sum_{n>=1} x^n/A(x)^n * (1+x)^n * Product_{k=1..n} (A(x) - x*(1+x)^(4*k-3)) / (A(x) - x*(1+x)^(4*k-1)), due to a q-series identity.
G.f.: 1+x = 1/(1 - q*x/(A(x) - q*(q^2-1)*x/(1 - q^5*x/(A(x) - q^3*(q^4-1)*x/(1 - q^9*x/(A(x) - q^5*(q^6-1)*x/(1 - q^13*x/(A(x) - q^7*(q^8-1)*x/(1 - ...))))))))), where q = (1+x), a continued fraction due to a partial elliptic theta function identity.

A303057 G.f. A(x) satisfies: A(x) = Sum_{n>=0} ((1+x)^n - 1)^n / A(x)^n.

Original entry on oeis.org

1, 1, 3, 21, 221, 3117, 54597, 1136127, 27293715, 742143113, 22512196673, 753402861159, 27571631761077, 1095346704175755, 46948527167219957, 2159638211148320085, 106129271000784614099, 5549226963359699829711, 307623817602110038648839, 18022345501064909362595723, 1112657716434830018636702797
Offset: 0

Views

Author

Paul D. Hanna, Apr 20 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 221*x^4 + 3117*x^5 + 54597*x^6 + 1136127*x^7 + 27293715*x^8 + 742143113*x^9 + 22512196673*x^10 + ...
such that
A(x) = 1 + ((1+x)-1)/A(x) + ((1+x)^2-1)^2/A(x)^2 + ((1+x)^3-1)^3/A(x)^3 + ((1+x)^4-1)^4/A(x)^4 + ((1+x)^5-1)^5/A(x)^5 + ((1+x)^6-1)^6/A(x)^6 + ...
also,
1 = 1/(A(x) + 1) + (1+x)/(A(x) + (1+x))^2 + (1+x)^4/(A(x) + (1+x)^2)^3 + (1+x)^9/(A(x) + (1+x)^3)^4 + (1+x)^16/(A(x) + (1+x)^4)^5 + (1+x)^25/(A(x) + (1+x)^5)^6 + (1+x)^36/(A(x) + (1+x)^6)^7 + ... + (1+x)^(n^2) / (A(x) + (1+x)^n)^(n+1) + ...

Links

Crossrefs

Cf. A303058.

Programs

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

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^n - 1)^n / A(x)^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n^2) / (A(x) + (1+x)^n)^(n+1). - Paul D. Hanna, Dec 13 2018
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.16108865386542881383... and c = 0.212154215724410476311... - Vaclav Kotesovec, Oct 06 2020

A318644 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (1+x)^(n*(n+1)/2) * x^n / A(x)^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 11, 32, 106, 376, 1433, 5782, 24574, 109393, 508026, 2453256, 12285347, 63656731, 340626704, 1879183856, 10672897341, 62323897482, 373748877678, 2299318074357, 14497472040378, 93599428822052, 618278575554155, 4175348680420942, 28806364292660618, 202899326988089615, 1458130019936912105, 10685096640964659318
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 11*x^6 + 32*x^7 + 106*x^8 + 376*x^9 + 1433*x^10 + 5782*x^11 + 24574*x^12 + 109393*x^13 + 508026*x^14 + ...
such that
A(x) = 1 + (1+x)*x/A(x) + (1+x)^3*x^2/A(x)^2 + (1+x)^6*x^3/A(x)^3 + (1+x)^10*x^4/A(x)^4 + (1+x)^15*x^5/A(x)^5 + (1+x)^21*x^6/A(x)^6 + (1+x)^28*x^7/A(x)^7 + ... + (1+x)^(n*(n+1)/2) * x^n / A(x)^n + ...
Also
1 + x = 1 + x/A(x) + (1+x)*x^2/A(x)^2 + (1+x)^3*x^3/A(x)^3 + (1+x)^6*x^4/A(x)^4 + (1+x)^10*x^5/A(x)^5 + (1+x)^15*x^6/A(x)^6 + (1+x)^21*x^7/A(x)^7 + ...

Links

Crossrefs

Cf. A320951, A303058.

Programs

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

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n * (1+x)^(n*(n+1)/2) / A(x)^n.
(2) 1 + x = Sum_{n>=0} x^n * (1+x)^(n*(n-1)/2) / A(x)^n.

A321607 G.f.: A(x) = Sum_{n>=0} x^n * (1+x)^(2*n^2) / A(x)^(2*n).

Original entry on oeis.org

1, 1, 1, 3, 11, 50, 294, 1833, 13093, 100456, 830541, 7313419, 68137388, 668327844, 6873073257, 73853969234, 826869429175, 9622413867975, 116144019094407, 1451337070767512, 18744666092043742, 249848115448337748, 3432213607060089249, 48532637291552045262, 705602120522021975822, 10536473807470843781980, 161442021516682347676435
Offset: 0

Views

Author

Paul D. Hanna, Nov 23 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 11*x^4 + 50*x^5 + 294*x^6 + 1833*x^7 + 13093*x^8 + 100456*x^9 + 830541*x^10 + 7313419*x^11 + 68137388*x^12 + ...
such that
A(x) = 1 + x*(1+x)^2/A(x)^2 + x^2*(1+x)^8/A(x)^4 + x^3*(1+x)^18/A(x)^6 + x^4*(1+x)^32/A(x)^8 + x^5*(1+x)^50/A(x)^10 + x^6*(1+x)^72/A(x)^12 + ...

Links

Crossrefs

Cf. A303058, A321608.

Programs

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

A321608 G.f.: A(x) = Sum_{n>=0} x^n * (1+x)^(2*n^2) / A(x)^n.

Original entry on oeis.org

1, 1, 2, 5, 22, 100, 581, 3716, 26352, 203664, 1688485, 14922599, 139579911, 1374581231, 14194823072, 153178320198, 1722363891798, 20130227813033, 244028318224686, 3062549991414408, 39723783490631858, 531726501000859625, 7335052105035725087, 104148717578191099492, 1520349684313678309811, 22793632534841442908565, 350622445923759834928352, 5528760266814203384425285
Offset: 0

Views

Author

Paul D. Hanna, Nov 23 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 22*x^4 + 100*x^5 + 581*x^6 + 3716*x^7 + 26352*x^8 + 203664*x^9 + 1688485*x^10 + 14922599*x^11 + 139579911*x^12 + ...
such that
A(x) = 1 + x*(1+x)^2/A(x) + x^2*(1+x)^8/A(x)^2 + x^3*(1+x)^18/A(x)^3 + x^4*(1+x)^32/A(x)^4 + x^5*(1+x)^50/A(x)^5 + x^6*(1+x)^72/A(x)^6 + ...

Links

Crossrefs

Cf. A303058, A321607.

Programs

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

A320951 G.f.: A(x) satisfies: A(x) = Sum_{n>=0} x^n * (1+x)^(n*(n+1)) / A(x)^n.

Original entry on oeis.org

1, 1, 2, 3, 9, 28, 110, 485, 2358, 12486, 70726, 425747, 2702837, 18004835, 125337381, 908737863, 6843536374, 53407750147, 431075414218, 3592384229312, 30862831600689, 272976843937138, 2482698463801148, 23192576636266041, 222310388884578760, 2184486850658804107, 21985733344615744620, 226455749821063728474, 2385331864619907236147, 25676170688883138634306, 282253492062060457638824
Offset: 0

Views

Author

Paul D. Hanna, Nov 20 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 9*x^4 + 28*x^5 + 110*x^6 + 485*x^7 + 2358*x^8 + 12486*x^9 + 70726*x^10 + 425747*x^11 + 2702837*x^12 + ...
such that
A(x) = 1 + x*(1+x)^2/A(x) + x^2*(1+x)^6/A(x)^2 + x^3*(1+x)^12/A(x)^3 + x^4*(1+x)^20/A(x)^4 + x^5*(1+x)^30/A(x)^5 + ...
Also
1 + x = 1 + x/A(x) + x^2*(1+x)^2/A(x)^2 + x^3*(1+x)^6/A(x)^3 + x^4*(1+x)^12/A(x)^4 + x^5*(1+x)^20/A(x)^5 + x^6*(1+x)^30/A(x)^6 + ...
RELATED SERIES.
Sum_{n>=0} x^n * (1+x)^(n^2) / A(x)^n = 1 + x + x^2 + x^3 + 3*x^4 + 8*x^5 + 32*x^6 + 135*x^7 + 649*x^8 + 3381*x^9 + 18894*x^10 + 112382*x^11 + 705174*x^12 + ...
A(A(x)-1) = 1 + x + 4*x^2 + 14*x^3 + 56*x^4 + 251*x^5 + 1239*x^6 + 6627*x^7 + 38112*x^8 + 233692*x^9 + 1517788*x^10 + 10384824*x^11 + ...
where A(A(x)-1) = Sum_{n>=0} (A(x)-1)^n * A(x)^(n*(n+1)) / A(A(x)-1)^n.

Links

Crossrefs

Cf. A318644, A303058.

Programs

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

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n * (1+x)^(n*(n+1)) / A(x)^n.
(2) 1 + x = Sum_{n>=0} x^n * (1+x)^(n*(n-1)) / A(x)^n.
(3) A(x) = Sum_{n>=0} (A(x)-1)^n * A(x)^(n*(n-1)) / A(A(x)-1)^n.

A320954 G.f. A(x) satisfies: 1/(1-x) = Sum_{n>=0} (1+x)^(n^2) * x^n / A(x)^n.

Original entry on oeis.org

1, 1, 2, 5, 14, 50, 200, 919, 4633, 25361, 148606, 923394, 6043996, 41447150, 296571213, 2206965193, 17034374165, 136066491764, 1122656493744, 9552206133005, 83695193972045, 754199756930791, 6981787930209535, 66327351641879318, 646031757787129761, 6445726513363688990, 65825739028009602120, 687540665329016479660
Offset: 0

Views

Author

Paul D. Hanna, Oct 25 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 50*x^5 + 200*x^6 + 919*x^7 + 4633*x^8 + 25361*x^9 + 148606*x^10 + 923394*x^11 + 6043996*x^12 + ...
such that
1/(1-x) = 1 + (1+x)*x/A(x) + (1+x)^4*x^2/A(x)^2 + (1+x)^9*x^3/A(x)^3 + (1+x)^16*x^4/A(x)^4 + (1+x)^25*x^5/A(x)^5 + (1+x)^36*x^6/A(x)^6 + ...

Links

Crossrefs

Cf. A303058, A301929.

Programs

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

Formula

Given g.f. A(x) then
(1) 1/(1-x) = 1/(1 - q*x/(A(x) - q*(q^2-1)*x/(1 - q^5*x/(A(x) - q^3*(q^4-1)*x/(1 - q^9*x/(A(x) - q^5*(q^6-1)*x/(1 - q^13*x/(A(x) - q^7*(q^8-1)*x/(1 - ...))))))))), where q = (1+x), a continued fraction due to a partial elliptic theta function identity.
(2) 1/(1-x) = Sum_{n>=0} x^n/A(x)^n * (1+x)^n * Product_{k=1..n} (A(x) - x*(1+x)^(4*k-3)) / (A(x) - x*(1+x)^(4*k-1)), due to a q-series identity.

A324614 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * (1 + n*x)^n / A(x)^(n+1).

Original entry on oeis.org

1, 1, 1, 2, 3, 9, 21, 76, 241, 962, 3687, 15930, 68993, 320025, 1511977, 7471685, 37780922, 197506241, 1056928087, 5810534182, 32667061545, 187952045908, 1104355482420, 6623724997302, 40514607315969, 252490521215350, 1602602016169781, 10349126940718990, 67984993381548943, 453846136553840921, 3078734565764856380, 21202631838742029002, 148238158399524358952, 1051257411796217414475
Offset: 0

Views

Author

Paul D. Hanna, Mar 19 2019

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 9*x^5 + 21*x^6 + 76*x^7 + 241*x^8 + 962*x^9 + 3687*x^10 + 15930*x^11 + 68993*x^12 + 320025*x^13 + 1511977*x^14 + ...
such that
1 = 1/A(x) + x*(1+x)/A(x)^2 + x^2*(1+2*x)^2/A(x)^3 + x^3*(1+3*x)^3/A(x)^4 + x^4*(1+4*x)^4/A(x)^5 + x^5*(1+5*x)^5/A(x)^6 + x^6*(1+6*x)^6/A(x)^7 + ...

Links

Crossrefs

Cf. A303058.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(n=0,#A, x^n*(1+n*x)^n/Ser(A)^(n+1)), #A-1););A[n+1]}
for(n=0,40, print1(a(n),", "))

A325155 G.f. A(x) satisfies: x = Sum_{n>=1} x^n * (1+x)^(n^2/2) / A(x)^(n/2).

Original entry on oeis.org

1, 3, 5, 7, 11, 21, 49, 133, 408, 1376, 5020, 19564, 80741, 350551, 1593066, 7547792, 37163568, 189662934, 1001046684, 5453972462, 30622950955, 176942133603, 1050773432990, 6405898358012, 40048848677954, 256521565555908, 1681897617101795, 11278819380424173, 77301464920178158, 541084956406886214, 3865540113371340736, 28167799470180443028, 209238063076396838375, 1583562040116769584091, 12204247180832799551059, 95731651337427271893873
Offset: 0

Views

Author

Paul D. Hanna, Apr 06 2019

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 5*x^2 + 7*x^3 + 11*x^4 + 21*x^5 + 49*x^6 + 133*x^7 + 408*x^8 + 1376*x^9 + 5020*x^10 + 19564*x^11 + 80741*x^12 + ...
such that
A(x) = 1 + x*(1+x)^(1/2)/A(x)^(1/2) + x^2*(1+x)^2/A(x) + x^3*(1+x)^(9/2)/A(x)^(3/2) + x^4*(1+x)^8/A(x)^2 + x^5*(1+x)^(25/2)/A(x)^(5/2) + x^6*(1+x)^18/A(x)^3 + x^7*(1+x)^(49/2)/A(x)^(7/2) + x^8*(1+x)^32/A(x)^4 + ...
Note that
sqrt(A(x))*sqrt(1+x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 11*x^6 + 32*x^7 + 106*x^8 + 376*x^9 + 1433*x^10 + 5782*x^11 + ... + A318644(n)*x^n + ...

Links

Crossrefs

Cf. A318644, A303058.

Programs

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

Formula

G.f. A(x) satisfies the following identities.
(1) 1 + x = Sum_{n>=0} x^n * (1+x)^(n^2/2) / A(x)^(n/2).
(2) 1 + x = 1/(1 - q*x/(sqrt(A(x)) - q*(q^2-1)*x/(1 - q^5*x/(sqrt(A(x)) - q^3*(q^4-1)*x/(1 - q^9*x/(sqrt(A(x)) - q^5*(q^6-1)*x/(1 - q^13*x/(sqrt(A(x)) - q^7*(q^8-1)*x/(1 - ...))))))))), where q = sqrt(1+x), a continued fraction due to a partial elliptic theta function identity.
(3) 1 + x = Sum_{n>=0} x^n * (1+x)^(n/2) / A(x)^(n/2) * Product_{k=1..n} (sqrt(A(x)) - x*sqrt(1+x)^(4*k-3)) / (sqrt(A(x)) - x*sqrt(1+x)^(4*k-1)), due to a q-series identity.
(4) A(x) = (1+x)*G(x)^2 where G(x) is the g.f. of A318644.
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