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.

A381818 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) / C(x))^2 ) )^(1/2), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 12, 97, 903, 9129, 97419, 1080058, 12319200, 143630575, 1704099034, 20507897766, 249734145622, 3071587654688, 38102046141882, 476138815310364, 5988435287060671, 75745116484532586, 962898676577135634, 12295850972794555196, 157649023155654522723, 2028662477759375282902
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A381817, A381819, A381820.
Cf. A364592, A381830, A381831.
Cf. A000108, A381772.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec((serreverse(x*((1-x)*2*x/(1-sqrt(1-4*x)))^2)/x)^(1/2))

Formula

G.f. A(x) satisfies A(x) = C(x*A(x)^2) / (1 - x*A(x)^2).
a(n) = Sum_{k=0..n} binomial(2*n+2*k+1,k) * binomial(3*n-k,n-k)/(2*n+2*k+1).

A381880 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / C(x) ), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 4, 27, 223, 2052, 20199, 208205, 2219149, 24261279, 270581313, 3066581130, 35216499786, 408919039968, 4792955710138, 56633333886618, 673881539636365, 8067939162382594, 97117925556632184, 1174721577627568371, 14270877151754826473, 174044527062280321368
Offset: 0

Views

Author

Seiichi Manyama, Mar 09 2025

Keywords

Crossrefs

Cf. A381817, A381879.
Cf. A000108, A381882.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3*2*x/(1-sqrt(1-4*x)))/x)

Formula

G.f. A(x) satisfies A(x) = C(x*A(x)) / (1 - x*A(x))^3.
a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(4*n-k+2,n-k)/(n+2*k+1).

A381879 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / C(x) ), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 3, 16, 106, 788, 6292, 52743, 457946, 4083328, 37174786, 344142192, 3229827900, 30661272627, 293907951057, 2840826401664, 27657352868946, 270968414904700, 2669604470832568, 26431802684789970, 262864480970961882, 2624640191306617088, 26301183967687772360
Offset: 0

Views

Author

Seiichi Manyama, Mar 09 2025

Keywords

Crossrefs

Cf. A381817, A381880.
Cf. A000108, A381881.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2*2*x/(1-sqrt(1-4*x)))/x)

Formula

G.f. A(x) satisfies A(x) = C(x*A(x)) / (1 - x*A(x))^2.
a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(3*n-k+1,n-k)/(n+2*k+1).

A381911 Expansion of (1/x) * Series_Reversion( x * (1-x) / B(x) ), where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 2, 9, 55, 394, 3102, 25969, 226891, 2045342, 18883205, 177640462, 1696658418, 16408796013, 160366113609, 1581329919636, 15713344659359, 157187582466527, 1581676730708500, 15998326150898211, 162571286470135097, 1658893916098102321, 16991130941208846890
Offset: 0

Views

Author

Seiichi Manyama, Mar 10 2025

Keywords

Crossrefs

Cf. A381912, A381913.
Cf. A381817, A381914.
Cf. A001764.

Programs

  • Mathematica
    Table[Sum[Binomial[n + 3*k + 1, k] * Binomial[2*n - k, n - k]/(n + 3*k + 1), {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 22 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+3*k+1, k)*binomial(2*n-k, n-k)/(n+3*k+1));

Formula

G.f. A(x) satisfies A(x) = B(x*A(x)) / (1 - x*A(x)).
a(n) = Sum_{k=0..n} binomial(n+3*k+1,k) * binomial(2*n-k,n-k)/(n+3*k+1).

A381819 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) / C(x))^3 ) )^(1/3), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 16, 177, 2271, 31731, 468614, 7195295, 113712012, 1837457589, 30220139048, 504212998955, 8513461623355, 145197727340337, 2497695979786842, 43285207907364178, 755005614380697735, 13244500528948104210, 233515959911770430972, 4135792046643993604967
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A381817, A381818, A381820.
Cf. A000108, A381773.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec((serreverse(x*((1-x)*2*x/(1-sqrt(1-4*x)))^3)/x)^(1/3))

Formula

G.f. A(x) satisfies A(x) = C(x*A(x)^3) / (1 - x*A(x)^3).
a(n) = Sum_{k=0..n} binomial(3*n+2*k+1,k) * binomial(4*n-k,n-k)/(3*n+2*k+1).

A381820 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) / C(x))^4 ) )^(1/4), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 20, 281, 4599, 82113, 1550993, 30473930, 616463800, 12753523628, 268586285058, 5738804673016, 124098812744140, 2710824280371114, 59728504549831296, 1325862161472193292, 29623682752417138511, 665679666998856945540, 15034747192791290846435
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A381817, A381818, A381819.
Cf. A000108, A381774.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec((serreverse(x*((1-x)*2*x/(1-sqrt(1-4*x)))^4)/x)^(1/4))

Formula

G.f. A(x) satisfies A(x) = C(x*A(x)^4) / (1 - x*A(x)^4).
a(n) = Sum_{k=0..n} binomial(4*n+2*k+1,k) * binomial(5*n-k,n-k)/(4*n+2*k+1).

A381914 Expansion of (1/x) * Series_Reversion( x * (1-x) / B(x) ), where B(x) is the g.f. of A002293.

Original entry on oeis.org

1, 2, 10, 72, 624, 6009, 61809, 664813, 7384613, 84045565, 974913510, 11483316680, 136974177209, 1651166320547, 20083352214058, 246168280262403, 3037682020219285, 37706043912831337, 470482875049515074, 5897864081341146065, 74243055437832292562, 938101296155866961124
Offset: 0

Views

Author

Seiichi Manyama, Mar 10 2025

Keywords

Crossrefs

Cf. A381915, A381916.
Cf. A381817, A381911.
Cf. A002293.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+4*k+1, k)*binomial(2*n-k, n-k)/(n+4*k+1));

Formula

G.f. A(x) satisfies A(x) = B(x*A(x)) / (1 - x*A(x)).
a(n) = Sum_{k=0..n} binomial(n+4*k+1,k) * binomial(2*n-k,n-k)/(n+4*k+1).

A381828 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) * (1-x+x^2))^2 ) )^(1/2).

Original entry on oeis.org

1, 2, 10, 65, 480, 3824, 32039, 278256, 2482578, 22617830, 209540672, 1968031520, 18696064179, 179332892186, 1734451272240, 16895744042472, 165621305486976, 1632518433458400, 16170959983623314, 160888256475481560, 1607061512154585046, 16110030923830784248
Offset: 0

Views

Author

Seiichi Manyama, Mar 08 2025

Keywords

Crossrefs

Cf. A188687, A381817, A381829.
Cf. A129442, A381831.
Cf. A000108, A368975.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec((serreverse(x*((1-x)*(1-x+x^2))^2)/x)^(1/2))

Formula

G.f. A(x) satisfies A(x) = C(x*A(x)) / (1 - x*A(x)^2), where C(x) is the g.f. of A000108.
a(n) = Sum_{k=0..n} binomial(2*n+k+1,k) * binomial(3*n-2*k,n-k)/(2*n+k+1).
D-finite with recurrence +432*n*(n-1)*(n-2)*(2*n+1)*(2*n-1)*(2*n-3)*(262261060139434887136491*n -880264534325728808928710)*a(n) +24*(n-1)*(n-2)*(2*n-1)*(2*n-3)*(9441398165019655936913676*n^3 -1563359509176097527827297363*n^2 +8122005300033248841454135898*n -10005843136737488906545668303)*a(n-1) -8*(n-2)*(2*n-3)*(26904862014415612504704360259*n^5 -439294650192331167438487778367*n^4 +2462557164881954865201862193560*n^3 -6116391863054255517662202621591*n^2 +6730597164009721987374566778403*n -2508886036978141982914230533400)*a(n-2) +2*(3280856375160701992555505608813*n^7 -60505233834440544774094319915261*n^6 +458650706405377012453301766859297*n^5 -1843996542698657351167896639498197*n^4 +4199211312282774397146042070543498*n^3 -5283107978583820687249123910721062*n^2 +3195330463869279708956264243293272*n -571272270914692694572799416918200)*a(n-3) +3*(-10499174187769013704183946812135*n^7 +189831332911960443054698384732480*n^6 -1395267797131742288585801071743534*n^5 +5221938509132769354051685228032464*n^4 -9839826026184653630837080778918103*n^3 +6229383740555425356174546560814416*n^2 +6216439623275682391743799709941612*n -8390747283534155728971424365124320)*a(n-4) -112*(7*n-31)*(7*n-32) *(2094251874056865218841652*n -5622141652266976856940223)*(7*n-29)*(7*n-26) *(7*n-30)*(7*n-27)*a(n-5)=0. - R. J. Mathar, Mar 10 2025

A381829 G.f. A(x) satisfies A(x) = C(x*A(x)) / (1 - x*A(x)^3), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 12, 97, 905, 9187, 98578, 1099980, 12636101, 148449436, 1775331503, 21541303494, 264533752068, 3281596216087, 41062196808517, 517655936768189, 6568539787903369, 83827401412072474, 1075254139150601581, 13855040994605807348, 179256835556387995412, 2327788724156294034612
Offset: 0

Views

Author

Seiichi Manyama, Mar 08 2025

Keywords

Crossrefs

Cf. A188687, A381817, A381828.
Cf. A000108, A381783.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(3*n+1, k)*binomial(4*n-3*k, n-k))/(3*n+1);

Formula

a(n) = (1/(3*n+1)) * Sum_{k=0..n} binomial(3*n+1,k) * binomial(4*n-3*k,n-k).
Showing 1-9 of 9 results.

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

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