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 14 results. Next

A365847 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1+x)^4 ).

Original entry on oeis.org

1, 8, 96, 1368, 21440, 356968, 6197408, 110947768, 2033381760, 37963483592, 719495148768, 13806129179928, 267693334199616, 5236670783633960, 103227182363423008, 2048451544990578552, 40888361539777714944, 820400146864231266184
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A066357, A365754, A365846, A365848.
Cf. A032349, A365622, A365843.
Cf. A260332.

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(4*n+k+3,k) * binomial(4*(n+1),n-k).
Conjecture: g.f.: B^4, where B is the g.f. of A260332.

A378238 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3*n+r+k,n)/(3*n+r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 14, 0, 1, 6, 32, 134, 0, 1, 8, 54, 324, 1482, 0, 1, 10, 80, 578, 3696, 17818, 0, 1, 12, 110, 904, 6810, 45316, 226214, 0, 1, 14, 144, 1310, 11008, 85278, 583152, 2984206, 0, 1, 16, 182, 1804, 16490, 140936, 1113854, 7769348, 40503890, 0
Offset: 0

Views

Author

Seiichi Manyama, Nov 20 2024

Keywords

Examples

			Square array begins:
  1,      1,      1,       1,       1,       1,       1, ...
  0,      2,      4,       6,       8,      10,      12, ...
  0,     14,     32,      54,      80,     110,     144, ...
  0,    134,    324,     578,     904,    1310,    1804, ...
  0,   1482,   3696,    6810,   11008,   16490,   23472, ...
  0,  17818,  45316,   85278,  140936,  216002,  314700, ...
  0, 226214, 583152, 1113854, 1870352, 2914790, 4320608, ...

Crossrefs

Columns k=0..3 give A000007, A144097, A371675, A365843.
T(n,n) gives 1/4 * A370102(n) for n > 0.
Cf. A033877, A071949, A378236, A378237, A378239, A378240.

Programs

  • PARI
    T(n, k, t=3, u=1) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(3/k) * (1 + A_k(x)^(1/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A144097.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-1,k+3) for n > 0.

A369264 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1+x^2)^3 ).

Original entry on oeis.org

1, 3, 18, 127, 993, 8268, 71888, 645087, 5929527, 55544315, 528319662, 5088941628, 49539243900, 486606281496, 4816930145376, 48005470976271, 481262635723491, 4850084768085465, 49107197378659262, 499298960719688343, 5095861705240094097
Offset: 0

Views

Author

Seiichi Manyama, Jan 18 2024

Keywords

Links

Crossrefs

Cf. A369262, A369263.
Cf. A365843, A369270.

Programs

  • Maple
    A369264 := proc(n)
    add(binomial(3*n+3,k) * binomial(4*n-2*k+2,n-2*k),k=0..floor(n/2)) ;
    %/(n+1) ;
end proc;
seq(A369264(n),n=0..70) ; # R. J. Mathar, Jan 25 2024
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3/(1+x^2)^3)/x)
  • PARI
    a(n, s=2, t=3, u=3) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(3*n+3,k) * binomial(4*n-2*k+2,n-2*k).
D-finite with recurrence +18*n*(3*n+2)*(2*n+3)*(3*n+1) *(2355222972296552964811*n -2353391098681877598217) *(n+1)*a(n) +3*n*(10232370941059360726949011*n^5 -6279411058144420889732231*n^4 +26515854213844281466097465*n^3 -21761373746876376187551525*n^2 -12108806260534489559295636*n +3394771165638813123794516)*a(n-1) +2*(-132629080888282243656059365*n^6 +156440924520330612537351287*n^5 -1546737637908414661531599805*n^4 +6858652031514251350543113065*n^3 -10688884261686986291502236950*n^2 +6884443241518652198616376568*n -1531720470240397832109679200)*a(n-2) +16*(-488032865226571716800174339*n^6 +5743512241166673419623793625*n^5 -28798925871340480498482300305*n^4 +76975939990931613139744649055*n^3 -114305622490237072905660442676*n^2 +89044784395613178550071941760*n -28430479725567026023998437760)*a(n-3) +384*(3*n-7) *(3*n-8)*(17416466042177225377415141*n^4 -183745766144088004186571330*n^3 +680994833213916542429809801*n^2 -1015881953145852406207817800*n +470197111913757817462248180)*a(n-4) +9216*(n-4)*(3*n-7)*(3*n-10) *(85246481204976073615097*n -71936955710157680798041)*(3*n-8) *(3*n-11)*a(n-5)=0. - R. J. Mathar, Jan 25 2024
a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^3 * (1+x^2)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024

A365842 Expansion of (1/x) * Series_Reversion( x*(1-x)^2/(1+x)^3 ).

Original entry on oeis.org

1, 5, 37, 325, 3141, 32261, 345605, 3818501, 43197445, 497868805, 5825331205, 69013667845, 826213203973, 9979713945605, 121472752156677, 1488482728148997, 18346810389299205, 227319830355640325, 2829629321065267205, 35369618935665131525, 443775430273133445125
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A007297, A365843, A365844, A365845.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(2*n+k+1, k)*binomial(3*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(2*n+k+1,k) * binomial(3*(n+1),n-k).
a(n) = (1/(n+1)) * [x^n] ( (1+x)^3 / (1-x)^2 )^(n+1). - Seiichi Manyama, Feb 17 2024

A369270 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1+x^3)^3 ).

Original entry on oeis.org

1, 3, 15, 94, 657, 4902, 38233, 307953, 2541831, 21386810, 182754162, 1581699162, 13836248406, 122139271098, 1086638457429, 9733419373534, 87707244737511, 794505072627735, 7231017033165776, 66089527981542462, 606340568510978940, 5582088822346925210
Offset: 0

Views

Author

Seiichi Manyama, Jan 18 2024

Keywords

Links

Crossrefs

Cf. A369268, A369269.
Cf. A365843, A369264.
Cf. A369232.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3/(1+x^3)^3)/x)
  • PARI
    a(n, s=3, t=3, u=3) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(3*n+3,k) * binomial(4*n-3*k+2,n-3*k).
a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^3 * (1+x^3)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024

A370098 a(n) = Sum_{k=0..n} binomial(3*n,k) * binomial(4*n-k-1,n-k).

Original entry on oeis.org

1, 6, 72, 978, 14016, 207006, 3116952, 47568618, 733189632, 11387193846, 177923724072, 2793666465090, 44042615547456, 696708049377294, 11053262513080440, 175800225426741978, 2802193910116429824, 44752001810800994022, 715924864099841086728
Offset: 0

Views

Author

Seiichi Manyama, Feb 10 2024

Keywords

Crossrefs

Cf. A091527, A370097.
Cf. A370099, A370102.
Cf. A365843.

Programs

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

Formula

a(n) = [x^n] ( (1+x)^3/(1-x)^3 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1-x)^3/(1+x)^3 ). See A365843.
From Seiichi Manyama, Aug 09 2025: (Start)
a(n) = [x^n] (1-x)^(n-1)/(1-2*x)^(3*n).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n,k) * binomial(n-1,n-k).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+k-1,k) * binomial(n-1,n-k). (End)

A365844 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1+x)^3 ).

Original entry on oeis.org

1, 7, 74, 931, 12894, 189798, 2913980, 46140347, 748022678, 12354604274, 207148525484, 3516699607022, 60328735646620, 1044182053141612, 18212018061261600, 319771572646888811, 5647677332549552870, 100266714048150595770, 1788366334642393259292
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A007297, A365842, A365843, A365845.

Programs

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

Formula

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

A365845 Expansion of (1/x) * Series_Reversion( x*(1-x)^5/(1+x)^3 ).

Original entry on oeis.org

1, 8, 97, 1400, 22243, 375584, 6614508, 120136984, 2234022775, 42322629960, 813939319697, 15849232257824, 311858145053076, 6191083938051840, 123852349440862504, 2494251111318893400, 50526944132627936127, 1028872756710478785560
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A007297, A365842, A365843, A365844.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*n+k+4, k)*binomial(3*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(5*n+k+4,k) * binomial(3*(n+1),n-k).

A365622 Expansion of (1/x) * Series_Reversion( x*(1-x)^5/(1+x)^5 ).

Original entry on oeis.org

1, 10, 150, 2670, 52250, 1086002, 23533790, 525825830, 12026993010, 280220428890, 6627397194022, 158692955007390, 3839595257256330, 93725694152075010, 2305406918530451950, 57085385625207424342, 1421808255906105290210
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A032349, A365843, A365847.
Cf. A363006.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*n+k+4, k)*binomial(5*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(5*n+k+4,k) * binomial(5*(n+1),n-k).
G.f.: B^5, where B is the g.f. of A363006.

A379245 G.f. A(x) satisfies A(x) = ( (1 + x*A(x)^2)/(1 - x*A(x)) )^3.

Original entry on oeis.org

1, 6, 72, 1100, 18984, 352608, 6879152, 139012368, 2884353888, 61091682368, 1315450042368, 28709737064064, 633684940733696, 14120739728984832, 317243001537462528, 7178031348934793472, 163423203504309020160, 3741114809852278047744
Offset: 0

Views

Author

Seiichi Manyama, Dec 18 2024

Keywords

Crossrefs

Cf. A365843, A379246.
Cf. A363380, A369215.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(3*n+3*k+3, k)*binomial(4*n+2*k+2, n-k)/(n+k+1));

Formula

G.f.: B(x)^3 where B(x) is the g.f. of A363380.
a(n) = Sum_{k=0..n} binomial(3*n+3*k+3,k) * binomial(4*n+2*k+2,n-k)/(n+k+1).
Showing 1-10 of 14 results. Next

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