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 1-6 of 6 results.

A371773 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-k+1,n-3*k).

Original entry on oeis.org

1, 3, 10, 36, 134, 507, 1937, 7449, 28783, 111623, 434130, 1692387, 6610292, 25861384, 101319095, 397428091, 1560588454, 6133768656, 24128550045, 94986663925, 374188128311, 1474980414870, 5817387549611, 22955930045826, 90629404431826, 357960414264163
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Links

Crossrefs

Cf. A371774, A371775, A371776.
Cf. A144904, A371758, A371777.

Programs

  • Mathematica
    Table[Sum[Binomial[2n-k+1,n-3k],{k,0,Floor[n/3]}],{n,0,30}] (* Harvey P. Dale, Sep 09 2024 *)
  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-k+1, n-3*k));

Formula

a(n) = [x^n] 1/(((1-x)^2-x^3) * (1-x)^n).
a(n) = binomial(2*(n-1), n-1)*hypergeom([1, (1-n)/3, (2-n)/3, 1-n/3], [1-n, 3/2-n, n], -27/4). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: n*(n^2 - 7)*a(n) = (9*n^3 - 2*n^2 - 79*n + 60)*a(n-1) - 2*(12*n^3 - 5*n^2 - 124*n + 150)*a(n-2) + (17*n^3 - 8*n^2 - 183*n + 240)*a(n-3) - 2*(2*n - 5)*(n^2 + 2*n - 6)*a(n-4).
a(n) ~ 2^(2*n+2) / sqrt(Pi*n). (End)

A371777 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n+2,n-3*k).

Original entry on oeis.org

1, 4, 15, 57, 220, 858, 3368, 13276, 52479, 207861, 824527, 3274395, 13015081, 51769813, 206045841, 820475513, 3268499356, 13025237058, 51922543076, 207034128448, 825713206746, 3293865399518, 13142007903586, 52443095356218, 209304385553096, 835459642193284
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Crossrefs

Cf. A371778, A371779, A371780.
Cf. A144904, A371758, A371773.
Cf. A032443.

Programs

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

Formula

a(n) = [x^n] 1/(((1-x)^3-x^3) * (1-x)^n).
a(n) = binomial(2*(n+1), n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [1+n/3, (4+n)/3, (5+n)/3], -1). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: n*a(n) = 3*(3*n-2)*a(n-1) - 6*(4*n-5)*a(n-2) + 8*(2*n-3)*a(n-3).
G.f.: (1 + sqrt(1-4*x))/(2*(1-x)*(1-4*x)).
a(n) ~ 2^(2*n+1)/3. (End)

A371770 a(n) = Sum_{k=0..floor(n/3)} binomial(3*n-3*k-1,n-3*k).

Original entry on oeis.org

1, 2, 10, 57, 338, 2057, 12741, 79914, 505954, 3226638, 20696685, 133382658, 862978221, 5601919325, 36467212610, 237974911737, 1556281907586, 10196788555859, 66921360130374, 439860632463462, 2895002186799453, 19077000179746293, 125849150650146714
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Links

Crossrefs

Cf. A005809, A165817, A183160.
Cf. A371758, A371771, A371772.

Programs

  • Maple
    f:= proc(n) local k; add(binomial(3*n-3*k-1,n-3*k),k=0..n/3) end proc:
map(f, [$0..30]); # Robert Israel, Feb 28 2025
  • PARI
    a(n) = sum(k=0, n\3, binomial(3*n-3*k-1, n-3*k));

Formula

a(n) = [x^n] 1/((1-x^3) * (1-x)^(2*n)).
a(n) = binomial(3*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [1/3-n, 2/3-n, 1-n], 1). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: 18*n*(2*n - 1)*(13*n - 22)*(37*n - 51)*a(n) = 3*(40885*n^4 - 165468*n^3 + 229373*n^2 - 125562*n + 22680)*a(n-1) - (40885*n^4 - 165468*n^3 + 229373*n^2 - 125562*n + 22680)*a(n-2) + 3*(3*n - 5)*(3*n - 4)*(13*n - 9)*(37*n - 14)*a(n-3).
a(n) ~ 3^(3*n + 5/2) / (13 * sqrt(Pi*n) * 2^(2*n+1)). (End)

A371771 a(n) = Sum_{k=0..floor(n/3)} binomial(4*n-3*k-1,n-3*k).

Original entry on oeis.org

1, 3, 21, 166, 1377, 11748, 102088, 898677, 7987305, 71517307, 644134026, 5829345492, 52964836184, 482846377185, 4414405051413, 40458397722306, 371605426607673, 3419639400458316, 31521758873514301, 291000881055737811, 2690082750919841442
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Links

Crossrefs

Cf. A005810, A147855, A262977.
Cf. A371758, A371770, A371772.

Programs

  • Maple
    f:= proc(n) local k; add(binomial(4*n-3*k-1,n-3*k),k=0..n/3) end proc:
map(f, [$0..30]); # Robert Israel, Feb 28 2025
  • PARI
    a(n) = sum(k=0, n\3, binomial(4*n-3*k-1, n-3*k));

Formula

a(n) = [x^n] 1/((1-x^3) * (1-x)^(3*n)).
a(n) = binomial(4*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [(1-4*n)/3, 2*(1-2*n)/3, 1-4*n/3], 1). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: 81*n*(3*n - 2)*(3*n - 1)*(9037*n^4 - 61391*n^3 + 154035*n^2 - 169317*n + 68836)*a(n) = 27*(2394805*n^7 - 19820156*n^6 + 66654684*n^5 - 117198990*n^4 + 115250735*n^3 - 62650734*n^2 + 17209736*n - 1814400)*a(n-1) - 3*(7021749*n^7 - 58192764*n^6 + 196050236*n^5 - 345531070*n^4 + 340849311*n^3 - 186035886*n^2 + 51353864*n - 5443200)*a(n-2) + 8*(2*n - 3)*(4*n - 9)*(4*n - 7)*(9037*n^4 - 25243*n^3 + 24084*n^2 - 9272*n + 1200)*a(n-3).
a(n) ~ 2^(8*n + 9/2) / (7 * sqrt(Pi*n) * 3^(3*n + 3/2)). (End)

A371772 a(n) = Sum_{k=0..floor(n/3)} binomial(5*n-3*k-1,n-3*k).

Original entry on oeis.org

1, 4, 36, 365, 3892, 42714, 477621, 5411109, 61901268, 713435333, 8271470666, 96361329024, 1127086021461, 13227336997645, 155680966681101, 1836862248992565, 21719923705450260, 257316706385394615, 3053599633736172765, 36292098436808314572, 431918050456887676362
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Crossrefs

Cf. A371758, A371770, A371771.
Cf. A371756.

Programs

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

Formula

a(n) = [x^n] 1/((1-x^3) * (1-x)^(4*n)).
a(n) = binomial(5*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [(1-5*n)/3, (2-5*n)/3, 1-5*n/3], 1). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: 72*n*(2*n - 1)*(4*n - 3)*(4*n - 1)*(899*n^2 - 2355*n + 1534)*a(n) = (25514519*n^6 - 117751221*n^5 + 212960873*n^4 - 191684487*n^3 + 89835824*n^2 - 20567076*n + 1769040)*a(n-1) - 5*(5*n - 7)*(5*n - 6)*(5*n - 4)*(5*n - 3)*(899*n^2 - 557*n + 78)*a(n-2).
a(n) ~ 5^(5*n + 5/2) / (31 * sqrt(Pi*n) * 2^(8*n + 3/2)). (End)

A371871 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-3*k-2,n-3*k).

Original entry on oeis.org

1, 0, 1, 5, 18, 66, 246, 924, 3493, 13277, 50697, 194327, 747319, 2882061, 11142027, 43167573, 167561586, 651513594, 2537041938, 9892847952, 38623197264, 150959213886, 590626854072, 2312979822738, 9065733950526, 35561306875380, 139595183125750
Offset: 0

Views

Author

Seiichi Manyama, Apr 10 2024

Keywords

Crossrefs

Cf. A360150, A371872, A371873.
Cf. A105872, A371758.

Programs

  • Maple
    A371871 := proc(n)
    1/(1-x^3)/(1-x)^(n-1) ;
    coeftayl(%,x=0,n) ;
end proc:
seq(A371871(n),n=0..60) ; # R. J. Mathar, Apr 22 2024
  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n-3*k-2, n-3*k));

Formula

a(n) = [x^n] 1/((1-x^3) * (1-x)^(n-1)).
D-finite with recurrence 9*n*a(n) +3*(-17*n+16)*a(n-1) +3*(21*n-50)*a(n-2) +(-17*n+16)*a(n-3) +10*(2*n-5)*a(n-4)=0. - R. J. Mathar, Apr 22 2024
Showing 1-6 of 6 results.

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