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 1-3 of 3 results.

A216698 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 6^k.

Original entry on oeis.org

1, 7, 85, 1351, 23281, 422527, 7951069, 153458935, 3018043777, 60225528727, 1215821974885, 24777776573095, 508935634491025, 10522995625652335, 218814097786515085, 4572338217781407031, 95953172529722919937, 2021236451413828339495, 42719661851354642952181
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 15 2012

Keywords

Links

Crossrefs

Cf. A000172 (x=1), A206178 (x=2), A206180 (x=3), A216483 (x=4), A216636 (x=5), A216696.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]^3*6^k, {k, 0, n}], {n, 0, 25}]
  • Sage
    A216698 = lambda n: hypergeometric([-n,-n,-n], [1,1], -6)
[Integer(A216698(n).n(100)) for n in (0..18)] # Peter Luschny, Sep 23 2014

Formula

General recurrecnce for Sum_{k=0..n} binomial(n,k)^3*x^k (this is case x=6): (n+3)^2*(3*n+4)*a(n+3) -(9*n^3+57*n^2+116*n+74)*(x+1)*a(n+2) +(3*n+5)*(3*n^2*(x^2-7*x+1)+11*n*(x^2-7*x+1)+9*x^2-66*x+9)*a(n+1) -(n+1)^2*(3*n+7)*(x+1)^3*a(n) = 0.
a(n) ~ (1+6^(1/3))^2/(2*2^(1/3)*3^(5/6)*Pi) * (1+6^(1/3))^(3*n)/n. - Vaclav Kotesovec, Sep 19 2012
G.f.: hypergeom([1/3, 2/3],[1],6*27*x^2/(1-7*x)^3)/(1-7*x). - Mark van Hoeij, May 02 2013
a(n) = hypergeom([-n,-n,-n],[1,1], -6). - Peter Luschny, Sep 23 2014

Extensions

Minor edits by Vaclav Kotesovec, Mar 31 2014

A216795 a(n) = sum_{k=0..n} binomial(n,k)^4 * 3^k.

Original entry on oeis.org

1, 4, 58, 1000, 19426, 412744, 9195796, 212836432, 5062716850, 123033947464, 3041489363188, 76243484446672, 1933564156575364, 49518970223489680, 1278877982692134568, 33269141292429734560, 870987510534775369810, 22930499187530338390600, 606700679139764282611540
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 16 2012

Keywords

Links

Crossrefs

Cf. A005260, A216696.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]^4*3^k, {k, 0, n}], {n, 0, 25}]

Formula

Recurrence: -(x-1)^4*(n+1)^3*(n+2)*(16*(4*x^2 + 17*x + 4)*n^4 + 184*(4*x^2 + 17*x + 4)*n^3 + (3137*x^2 + 13351*x + 3137)*n^2 + (5867*x^2 + 25051*x + 5867)*n + 4061*x^2 + 17438*x + 4061)*a(n) + (n+2)*(64*(4*x^5 + 141*x^4 + 655*x^3 + 655*x^2 + 141*x + 4)*n^7 + 1024*(4*x^5 + 141*x^4 + 655*x^3 + 655*x^2 + 141*x + 4)*n^6 + 4*(6857*x^5 + 242368*x^4 + 1126775*x^3 + 1126775*x^2 + 242368*x + 6857)*n^5 + 8*(12439*x^5 + 442336*x^4 + 2059985*x^3 + 2059985*x^2 + 442336*x + 12439)*n^4 + (211031*x^5 + 7579744*x^4 + 35400065*x^3 + 35400065*x^2 + 7579744*x + 211031)*n^3 + (261344*x^5 + 9524206*x^4 + 44667470*x^3 + 44667470*x^2 + 9524206*x + 261344)*n^2 + (174888*x^5 + 6498997*x^4 + 30655175*x^3 + 30655175*x^2 + 6498997*x + 174888)*n + 15*(3251*x^5 + 123835*x^4 + 588594*x^3 + 588594*x^2 + 123835*x + 3251))*a(n+1) -(32*(12*x^4-197*x^3-1030*x^2-197*x + 12)*n^8 + 624*(12*x^4-197*x^3-1030*x^2-197*x + 12)*n^7 + 10*(6295*x^4-103673*x^3-542004*x^2-103673*x + 6295)*n^6 + 4*(74418*x^4-1233343*x^3-6448430*x^2-1233343*x + 74418)*n^5 + (864893*x^4-14467663*x^3-75685660*x^2-14467663*x + 864893)*n^4 + 20*(78938*x^4-1336491*x^3-7002327*x^2-1336491*x + 78938)*n^3 + (1764932*x^4-30321697*x^3-159367410*x^2-30321697*x + 1764932)*n^2 + (1102551*x^4-19262286*x^3-101826010*x^2-19262286*x + 1102551)*n + 10*(29405*x^4-523232*x^3-2793306*x^2-523232*x + 29405))*a(n+2) + (n+3)*(64*(4*x^3 + 21*x^2 + 21*x + 4)*n^7 + 1152*(4*x^3 + 21*x^2 + 21*x + 4)*n^6 + 12*(2899*x^3 + 15226*x^2 + 15226*x + 2899)*n^5 + 4*(35609*x^3 + 187226*x^2 + 187226*x + 35609)*n^4 + (340693*x^3 + 1795162*x^2 + 1795162*x + 340693)*n^3 + (474743*x^3 + 2511132*x^2 + 2511132*x + 474743)*n^2 + (355831*x^3 + 1894439*x^2 + 1894439*x + 355831)*n + 10*(11039*x^3 + 59401*x^2 + 59401*x + 11039))*a(n+3) -(n+3)*(n+4)^3*(16*(4*x^2 + 17*x + 4)*n^4 + 120*(4*x^2 + 17*x + 4)*n^3 + (1313*x^2 + 5599*x + 1313)*n^2 + 15*(103*x^2 + 443*x + 103)*n + 659*x^2 + 2882*x + 659)*a(n+4) = 0, this is case x=3.
a(n) ~ (1+3^(1/4))^3/(4*sqrt(2)*3^(3/8)*Pi^(3/2)) * (1+3^(1/4))^(4*n)/n^(3/2). - Vaclav Kotesovec, Sep 19 2012
Generally, Sum_{k=0..n} binomial(n,k)^p*x^k is asymptotic a(n) ~ (1+x^(1/p))^(p*n+p-1)/sqrt((2*pi*n)^(p-1)*p*x^(1-1/p)). This is case p=4, x=3. - Vaclav Kotesovec, Sep 19 2012

Extensions

Minor edits by Vaclav Kotesovec, Mar 31 2014

A336203 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} 2^j * binomial(n,j)^k.

Original entry on oeis.org

1, 1, 3, 1, 3, 7, 1, 3, 9, 15, 1, 3, 13, 27, 31, 1, 3, 21, 63, 81, 63, 1, 3, 37, 171, 321, 243, 127, 1, 3, 69, 495, 1521, 1683, 729, 255, 1, 3, 133, 1467, 7761, 14283, 8989, 2187, 511, 1, 3, 261, 4383, 41361, 131283, 138909, 48639, 6561, 1023, 1, 3, 517, 13131, 227601, 1256283, 2336629, 1385163, 265729, 19683, 2047
Offset: 0

Views

Author

Seiichi Manyama, Jul 11 2020

Keywords

Comments

Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) - 2 * Product_{j=1..k} x_j) for k>0.

Examples

			Square array begins:
   1,   1,    1,     1,      1,       1, ...
   3,   3,    3,     3,      3,       3, ...
   7,   9,   13,    21,     37,      69, ...
  15,  27,   63,   171,    495,    1467, ...
  31,  81,  321,  1521,   7761,   41361, ...
  63, 243, 1683, 14283, 131283, 1256283, ...

Crossrefs

Columns k=0-4 give: A000225(n+1), A000244, A001850, A206178, A216696.
Main diagonal gives A336204.
Cf. A309010.

Programs

  • Mathematica
    T[n_, k_] := Sum[2^j * Binomial[n, j]^k, {j, 0, n}]; Table[T[k, n-k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)
Showing 1-3 of 3 results.

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