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.

A301435 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 3*x*A(x) )^n / 2^(n+1).

Original entry on oeis.org

1, 5, 85, 2413, 92501, 4394663, 246960721, 15952488893, 1161947365721, 94123508334877, 8390631582459161, 816285612080072183, 86069411025655759073, 9778818588385117669485, 1191176369495005591666205, 154886342347657508336231809, 21414816209632043592416524165, 3137473307880710686085483679771, 485584927860050612832028930482597, 79169341280742628145184619086229089
Offset: 0

Views

Author

Paul D. Hanna, Mar 24 2018

Keywords

Examples

			G.f.: A(x) = 1 + 5*x + 85*x^2 + 2413*x^3 + 92501*x^4 + 4394663*x^5 + 246960721*x^6 + 15952488893*x^7 + 1161947365721*x^8 + 94123508334877*x^9 + ...
such that
1 = 1/2  +  ((1+x) - 3*x*A(x))/2^2  +  ((1+x)^2 - 3*x*A(x))^2/2^3  +  ((1+x)^3 - 3*x*A(x))^3/2^4  +  ((1+x)^4 - 3*x*A(x))^4/2^5  +  ((1+x)^5 - 3*x*A(x))^5/2^6  + ...
Also,
1 = 1/(2 + 3*x*A(x)) + (1+x)/(2 + 3*x*A(x)*(1+x))^2 + (1+x)^4/(2 + 3*x*A(x)*(1+x)^2)^3 + (1+x)^9/(2 + 3*x*A(x)*(1+x)^3)^4 + (1+x)^16/(2 + 3*x*A(x)*(1+x)^4)^5 + ...
RELATED SERIES.
Let R(k,x) = Sum_{n>=0} binomial(n+k,k) * (1+x)^(n*(n+k)) / 2^(n+k+1)
then
1 = R(0,x) - 3*x*A(x)*R(1,x) + 3^2*x^2*A(x)^2*R(2,x) - 3^3*x^3*A(x)^3*R(3,x) + 3^4*x^4*A(x)^4*R(4,x) - 3^5*x^5*A(x)^5*R(5,x) + ...
The table of coefficients in R(k,x) begins:
k=0: [1, 3, 36, 744, 21606, 807912, 36948912, 1997801520, ...];
k=1: [1, 10, 197, 5600, 206880, 9387864, 504836996, 31376330400, ...];
k=2: [1, 21, 621, 23447, 1078980, 58590504, 3667676768, ...];
k=3: [1, 36, 1494, 72516, 4075569, 261336096, 18861815280, ...];
k=4: [1, 55, 3050, 185190, 12492745, 934629539, 77091424200, ...];
k=5: [1, 78, 5571, 413764, 33004131, 2850142590, 266518090901, ...];
k=6: [1, 105, 9387, 837165, 77946645, 7696470411, 810015165897, ...];
k=7: [1, 136, 14876, 1568632, 168591350, 18874524760, 2221139481932, ...]; ...

Links

Crossrefs

Cf. A303288, A301436, A323314, A323315, A323316, A323317, A323318, A323319, A323320, A323321.

Programs

  • Mathematica
    nmax = 19;
nmax2 = 300 (* = empirical sum terms *);
sol = {a[0] -> 1};
A[x_] = Sum[a[n] x^n, {n, 0, nmax}];
Do[A[x] = A[x] /. sol; s = 1-Sum[((1+x)^n - 3x A[x] + O[x]^(k+1))^n / 2^(n+1), {n, 0, nmax2}] /. sol; c = SeriesCoefficient[s, {x, 0, k}]; sol = sol ~Join~ Solve[c == 0][[1]] /. HoldPattern[a[n_] -> an_] :> (a[n] -> Round[an]), {k, 2, nmax+1}];
a /@ Range[0, nmax] /. sol (* Jean-François Alcover, Nov 05 2019 *)

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( (1+x)^n - 3*x*A(x) )^n / 2^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n^2) / (2 + 3*x*A(x)*(1+x)^n)^(n+1). - Paul D. Hanna, Jan 10 2019
(3) 1 = Sum_{k>=0} (-3*x)^k * A(x)^k * Sum_{n>=0} C(n+k,k) * (1+x)^(n*(n+k)) / 2^(n+k+1).
(4) 1 = Sum_{n>=0} Sum_{k=0..n} C(n,k) * (1+x)^(n*(n-k)) / 2^(n+1) * (-3*x)^k * A(x)^k.

A303288 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 5*x*A(x) )^n * 2^n / 3^(n+1).

Original entry on oeis.org

1, 14, 688, 56738, 6347176, 881241656, 144796770004, 27351977086556, 5826096152426212, 1380051673281134312, 359720002818554238352, 102317793242070983628176, 31540355035889303797419616, 10475792506313141986771902704, 3730248479020018845292570520560, 1417811189172027111629537752756520, 572992474515466430293335350543824096
Offset: 0

Views

Author

Paul D. Hanna, Apr 23 2018

Keywords

Examples

			G.f.: A(x) = 1 + 14*x + 688*x^2 + 56738*x^3 + 6347176*x^4 + 881241656*x^5 + 144796770004*x^6 + 27351977086556*x^7 + 5826096152426212*x^8 + ...
such that
1 = 1/3 + 2*((1+x) - 5*x*A(x))/3^2 + 2^2*((1+x)^2 - 5*x*A(x))^2/3^3 + 2^3*((1+x)^3 - 5*x*A(x))^3/3^4 + 2^4*((1+x)^4 - 5*x*A(x))^4/3^5 + 2^5*((1+x)^5 - 5*x*A(x))^5/3^6 + ...
also,
1 = 1/(3 + 10*x*A(x)) + 2*(1+x)/(3 + 10*x*A(x))^2 + 2^2*(1+x)^4/(3 + 10*x*A(x))^3 + 2^3*(1+x)^9/(3 + 10*x*A(x))^4 + 2^4*(1+x)^16/(3 + 10*x*A(x))^5 + 2^5*(1+x)^25/(3 + 10*x*A(x))^6 + ...

Links

Crossrefs

Cf. A301435, A303291, A323314, A323315, A323316, A323317, A323318, A323319, A323320, A323321.

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} 2^n * ( (1+x)^n - 5*x*A(x) )^n / 3^(n+1).
(2) 1 = Sum_{n>=0} 2^n * (1+x)^(n^2) / (3 + 10*x*A(x)*(1+x)^n)^(n+1). - Paul D. Hanna, Jan 10 2019
(3) 1 = Sum_{k>=0} (-5*x)^k * A(x)^k * Sum_{n>=0} C(n+k,k) * (1+x)^(n*(n+k)) * 2^(n+k) / 3^(n+k+1).
(4) 1 = Sum_{n>=0} Sum_{k=0..n} C(n,k) * (1+x)^(n*(n-k)) * 2^n / 3^(n+1) * (-5*x)^k * A(x)^k.

A323315 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 9*x*A(x) )^n * 4^n / 5^(n+1).

Original entry on oeis.org

1, 44, 7096, 1926724, 711117536, 325957899584, 176862173366416, 110333447177205584, 77614355506352291216, 60715204091160869904064, 52262738608604757586146176, 49089829530793158665498530304, 49969859760169581295921965453056, 54804935053668330661788935789639424, 64439695005477056297527256416094395136, 80874250846078911532650120181881418467904
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2019

Keywords

Examples

			G.f.: A(x) = 1 + 44*x + 7096*x^2 + 1926724*x^3 + 711117536*x^4 + 325957899584*x^5 + 176862173366416*x^6 + 110333447177205584*x^7 + ...
such that
1 = 1/5 + ((1+x) - 9*x*A(x))*4/5^2 + ((1+x)^2 - 9*x*A(x))^2*4^2/5^3 + ((1+x)^3 - 9*x*A(x))^3*4^3/5^4 + ((1+x)^4 - 9*x*A(x))^4*4^4/5^5 + ...
Also,
1 = 1/(5 + 36*x*A(x)) + (1+x)*4/(5 + 36*x*A(x)*(1+x))^2 + (1+x)^4*4^2/(5 + 36*x*A(x)*(1+x)^2)^3 + (1+x)^9*4^3/(5 + 36*x*A(x)*(1+x)^3)^4 + ...

Crossrefs

Cf. A301435, A303288, A323314, A323316, A323317, A323318, A323319, A323320, A323321.

Programs

  • PARI
    \p120
{A=vector(1); A[1]=1; for(i=1,20, A = concat(A,0);
A[#A] = round( Vec( sum(n=0,1000, ( (1+x +x*O(x^#A))^n - 9*x*Ser(A) )^n * 4^n/5^(n+1)*1.)/36 ) )[#A+1]); A}

Formula

G.f. A(x) satisfies the following identities.
(1) 1 = Sum_{n>=0} ( (1+x)^n - 9*x*A(x) )^n * 4^n / 5^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n^2) * 4^n / (5 + 36*x*A(x)*(1+x)^n)^(n+1).

A323316 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 11*x*A(x) )^n * 5^n / 6^(n+1).

Original entry on oeis.org

1, 65, 15685, 6376505, 3524871325, 2420187902975, 1967093055766825, 1838251199473028225, 1937082794808580188025, 2269921874941072916242625, 2926922052137279952439869625, 4118264067683762888405147993375, 6279611163775388892921689107812625, 10316794138820163374949788420225125625, 18170957626950430345183391610737313950125, 34161178486729901360568404660435153779920125
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2019

Keywords

Examples

			G.f.: A(x) = 1 + 65*x + 15685*x^2 + 6376505*x^3 + 3524871325*x^4 + 2420187902975*x^5 + 1967093055766825*x^6 + 1838251199473028225*x^7 + ...
such that
1 = 1/6 + ((1+x) - 11*x*A(x))*5/6^2 + ((1+x)^2 - 11*x*A(x))^2*5^2/6^3 + ((1+x)^3 - 11*x*A(x))^3*5^3/6^4 + ((1+x)^4 - 11*x*A(x))^4*5^4/6^5 + ...
Also,
1 = 1/(6 + 55*x*A(x)) + (1+x)*5/(6 + 55*x*A(x)*(1+x))^2 + (1+x)^4*5^2/(6 + 55*x*A(x)*(1+x)^2)^3 + (1+x)^9*5^3/(6 + 55*x*A(x)*(1+x)^3)^4 + ...

Crossrefs

Cf. A301435, A303288, A323314, A323315, A323317, A323318, A323319, A323320, A323321.

Programs

  • PARI
    \p120
{A=vector(1); A[1]=1; for(i=1,20, A = concat(A,0);
A[#A] = round( Vec( sum(n=0,1200, ( (1+x +x*O(x^#A))^n - 11*x*Ser(A) )^n * 5^n/6^(n+1)*1.)/55 ) )[#A+1]); A}

Formula

G.f. A(x) satisfies the following identities.
(1) 1 = Sum_{n>=0} ( (1+x)^n - 11*x*A(x) )^n * 5^n / 6^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n^2) * 5^n / (6 + 55*x*A(x)*(1+x)^n)^(n+1).

A323317 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 13*x*A(x) )^n * 6^n / 7^(n+1).

Original entry on oeis.org

1, 90, 30360, 17260998, 13346871336, 12819352461768, 14575804541933076, 19054882926950474988, 28089490655708754330276, 46046879475849529578435672, 83060213421430745855381951856, 163488644041366509740041070551248, 348735916991281119541339971532867488, 801490465035993025759896936239032263600, 1974787497208210693752899355242321943894000, 5193543503462268857667579481311302800804588450
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2019

Keywords

Examples

			G.f.: A(x) = 1 + 90*x + 30360*x^2 + 17260998*x^3 + 13346871336*x^4 + 12819352461768*x^5 + 14575804541933076*x^6 + 19054882926950474988*x^7 + ...
such that
1 = 1/7 + ((1+x) - 13*x*A(x))*6/7^2 + ((1+x)^2 - 13*x*A(x))^2*6^2/7^3 + ((1+x)^3 - 13*x*A(x))^3*6^3/7^4 + ((1+x)^4 - 13*x*A(x))^4*6^4/7^5 + ...
Also,
1 = 1/(7 + 78*x*A(x)) + (1+x)*6/(7 + 78*x*A(x)*(1+x))^2 + (1+x)^4*6^2/(7 + 78*x*A(x)*(1+x)^2)^3 + (1+x)^9*6^3/(7 + 78*x*A(x)*(1+x)^3)^4 + ...

Crossrefs

Cf. A301435, A303288, A323314, A323315, A323316, A323318, A323319, A323320, A323321.

Programs

  • PARI
    \p120
{A=vector(1); A[1]=1; for(i=1,20, A = concat(A,0);
A[#A] = round( Vec( sum(n=0,1200, ( (1+x +x*O(x^#A))^n - 13*x*Ser(A) )^n * 6^n/7^(n+1)*1.)/78 ) )[#A+1]); A}

Formula

G.f. A(x) satisfies the following identities.
(1) 1 = Sum_{n>=0} ( (1+x)^n - 13*x*A(x) )^n * 6^n / 7^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n^2) * 6^n / (7 + 78*x*A(x)*(1+x)^n)^(n+1).

A323318 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 15*x*A(x) )^n * 7^n / 8^(n+1).

Original entry on oeis.org

1, 119, 53473, 40508503, 41741036561, 53428266259151, 80958982980046129, 141048455946249441191, 277099512762218200167617, 605370915659340921493495687, 1455268739680049030318517763457, 3817384299846582450604884256739951, 10851817459553385455156107655677525601, 33237713019302068995081812342685224005719, 109138923772997447194531532072327732171764385
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2019

Keywords

Examples

			G.f.: A(x) = 1 + 119*x + 53473*x^2 + 40508503*x^3 + 41741036561*x^4 + 53428266259151*x^5 + 80958982980046129*x^6 + 141048455946249441191*x^7 + ...
such that
1 = 1/8 + ((1+x) - 15*x*A(x))*7/8^2 + ((1+x)^2 - 15*x*A(x))^2*7^2/8^3 + ((1+x)^3 - 15*x*A(x))^3*7^3/8^4 + ((1+x)^4 - 15*x*A(x))^4*7^4/8^5 + ...
Also,
1 = 1/(8 + 105*x*A(x)) + (1+x)*7/(8 + 105*x*A(x)*(1+x))^2 + (1+x)^4*7^2/(8 + 105*x*A(x)*(1+x)^2)^3 + (1+x)^9*7^3/(8 + 105*x*A(x)*(1+x)^3)^4 + ...

Crossrefs

Cf. A301435, A303288, A323314, A323315, A323316, A323317, A323319, A323320, A323321.

Programs

  • PARI
    \p120
{A=vector(1); A[1]=1; for(i=1,20, A = concat(A,0);
A[#A] = round( Vec( sum(n=0, 2000, ( (1+x +x*O(x^#A))^n - 15*x*Ser(A) )^n * 7^n/8^(n+1)*1.)/105 ) )[#A+1]); A}

Formula

G.f. A(x) satisfies the following identities.
(1) 1 = Sum_{n>=0} ( (1+x)^n - 15*x*A(x) )^n * 7^n / 8^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n^2) * 7^n / (8 + 105*x*A(x)*(1+x)^n)^(n+1).

A323319 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 17*x*A(x) )^n * 8^n / 9^(n+1).

Original entry on oeis.org

1, 152, 87760, 85439240, 113151839104, 186152435786240, 362548564958149696, 811847325733606058048, 2049967057729258844550208, 5756221555712461523954507264, 17785396518936498493080842349568, 59963943179536216027803213130483712, 219093913413498532617018883655015864320, 862506026576114820987041351988191302565888, 3640101913203153345185251232178995247004487680, 16397805545827151302219567488776238270687543337472
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2019

Keywords

Examples

			G.f.: A(x) = 1 + 152*x + 87760*x^2 + 85439240*x^3 + 113151839104*x^4 + 186152435786240*x^5 + 362548564958149696*x^6 + 811847325733606058048*x^7 + ...
such that
1 = 1/9 + ((1+x) - 17*x*A(x))*8/9^2 + ((1+x)^2 - 17*x*A(x))^2*8^2/9^3 + ((1+x)^3 - 17*x*A(x))^3*8^3/9^4 + ((1+x)^4 - 17*x*A(x))^4*8^4/9^5 + ...
Also,
1 = 1/(9 + 136*x*A(x)) + (1+x)*8/(9 + 136*x*A(x)*(1+x))^2 + (1+x)^4*8^2/(9 + 136*x*A(x)*(1+x)^2)^3 + (1+x)^9*8^3/(9 + 136*x*A(x)*(1+x)^3)^4 + ...

Crossrefs

Cf. A301435, A303288, A323314, A323315, A323316, A323317, A323318, A323320, A323321.

Programs

  • PARI
    \p120
{A=vector(1); A[1]=1; for(i=1,20, A = concat(A,0);
A[#A] = round( Vec( sum(n=0, 2200, ( (1+x +x*O(x^#A))^n - 17*x*Ser(A) )^n * 8^n/9^(n+1)*1.)/136 ) )[#A+1]); A}

Formula

G.f. A(x) satisfies the following identities.
(1) 1 = Sum_{n>=0} ( (1+x)^n - 17*x*A(x) )^n * 8^n / 9^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n^2) * 8^n / (9 + 136*x*A(x)*(1+x)^n)^(n+1).

A323320 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 19*x*A(x) )^n * 9^n / 10^(n+1).

Original entry on oeis.org

1, 189, 136341, 165866949, 274513563621, 564389814803319, 1373687351977035681, 3844220718032111632869, 12130905677234774784280281, 42569255610714760893622565829, 164374338314267349285576891426201, 692583662656534583930262265650693159, 3162450027762781275258550249673787013761, 15558457725978409248029649314240444710279749, 82059484588450416190385956503916602281112899421
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2019

Keywords

Examples

			G.f.: A(x) = 1 + 189*x + 136341*x^2 + 165866949*x^3 + 274513563621*x^4 + 564389814803319*x^5 + 1373687351977035681*x^6 + 3844220718032111632869*x^7 + ...
such that
1 = 1/10 + ((1+x) - 19*x*A(x))*9/10^2 + ((1+x)^2 - 19*x*A(x))^2*9^2/10^3 + ((1+x)^3 - 19*x*A(x))^3*9^3/10^4 + ((1+x)^4 - 19*x*A(x))^4*9^4/10^5 + ...
Also,
1 = 1/(10 + 171*x*A(x)) + (1+x)*9/(10 + 171*x*A(x)*(1+x))^2 + (1+x)^4*9^2/(10 + 171*x*A(x)*(1+x)^2)^3 + (1+x)^9*9^3/(10 + 171*x*A(x)*(1+x)^3)^4 + ...

Crossrefs

Cf. A301435, A303288, A323314, A323315, A323316, A323317, A323318, A323319, A323321.

Programs

  • PARI
    \p120
{A=vector(1); A[1]=1; for(i=1,20, A = concat(A,0);
A[#A] = round( Vec( sum(n=0, 2400, ( (1+x +x*O(x^#A))^n - 19*x*Ser(A) )^n * 9^n/10^(n+1)*1.)/171 ) )[#A+1]); A}

Formula

G.f. A(x) satisfies the following identities.
(1) 1 = Sum_{n>=0} ( (1+x)^n - 19*x*A(x) )^n * 9^n / 10^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n^2) * 9^n / (10 + 171*x*A(x)*(1+x)^n)^(n+1).

A323321 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 21*x*A(x) )^n * 10^n / 11^(n+1).

Original entry on oeis.org

1, 230, 202720, 301356010, 609471837800, 1531246512757400, 4554410421462541300, 15575127764655971159900, 60061431635374301463364100, 257559473604548074955131621000, 1215330203862647096788767608162000, 6257647362127152791857282855542122000, 34917317338173226632480770480063290796000, 209923913089512941533199772776123546222790000, 1353013627656130991705167318085125179145490486000
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2019

Keywords

Examples

			G.f.: A(x) = 1 + 230*x + 202720*x^2 + 301356010*x^3 + 609471837800*x^4 + 1531246512757400*x^5 + 4554410421462541300*x^6 + 15575127764655971159900*x^7 + ...
such that
1 = 1/11 + ((1+x) - 21*x*A(x))*10/11^2 + ((1+x)^2 - 21*x*A(x))^2*10^2/11^3 + ((1+x)^3 - 21*x*A(x))^3*10^3/11^4 + ((1+x)^4 - 21*x*A(x))^4*10^4/11^5 + ...
Also,
1 = 1/(11 + 210*x*A(x)) + (1+x)*10/(11 + 210*x*A(x)*(1+x))^2 + (1+x)^4*10^2/(11 + 210*x*A(x)*(1+x)^2)^3 + (1+x)^9*10^3/(11 + 210*x*A(x)*(1+x)^3)^4 + ...

Crossrefs

Cf. A301435, A303288, A323314, A323315, A323316, A323317, A323318, A323319, A323320.

Programs

  • PARI
    \p120
{A=vector(1); A[1]=1; for(i=1,20, A = concat(A,0);
A[#A] = round( Vec( sum(n=0, 3000, ( (1+x +x*O(x^#A))^n - 21*x*Ser(A) )^n * 10^n/11^(n+1)*1.)/210 ) )[#A+1]); A}

Formula

G.f. A(x) satisfies the following identities.
(1) 1 = Sum_{n>=0} ( (1+x)^n - 21*x*A(x) )^n * 10^n / 11^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n^2) * 10^n / (11 + 210*x*A(x)*(1+x)^n)^(n+1).
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