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.

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A124995 a(n) is the constant term in expansion of Product_{ k = 1..n } (x^k + 1/x^k)^3.

Original entry on oeis.org

1, 0, 0, 62, 332, 0, 0, 80006, 531524, 0, 0, 173607568, 1226700784, 0, 0, 455805857978, 3321800235936, 0, 0, 1325490660318216, 9841000101286172, 0, 0, 4108826483323392880, 30886378286619335592, 0, 0, 13306426381421174346512, 100916492010297213463566
Offset: 0

Views

Author

N. J. A. Sloane, Jul 12 2008

Keywords

Comments

From Robert Israel, Nov 09 2017: (Start)
a(n) is the coefficient of x^(3*n*(n+1)/2) in Product_{k=0..n} (x^(2*k)+1)^3.
a(n) = 0 if n == 1 or 2 (mod 4). (End)

Links

Crossrefs

For constant term in expansion of Product_{ k = 1..n } (x^k + 1/x^k)^q for other values of q see A063865, A047653, A124996.

Programs

  • Maple
    seq(coeff(mul(x^k+1/x^k,k=1..n)^3,x,0),n=0..50); # Robert Israel, Nov 09 2017
  • PARI
    a(n) = polcoef(prod(k=1, n, (x^k + 1/x^k)^3), 0); \\ Michel Marcus, Jan 07 2021

A124996 a(n) is the constant term in the expansion of Product_{k=1..n} (x^k + 1/x^k)^4.

Original entry on oeis.org

1, 6, 44, 426, 4658, 55260, 689508, 8914872, 118374410, 1604658420, 22115171280, 308940507202, 4364729023812, 62256518307724, 895294865045296, 12966655239260890, 188967013096930258, 2769003814616561636, 40773380119956434784, 603008173331642200144
Offset: 0

Views

Author

N. J. A. Sloane, Jul 12 2008

Keywords

Links

Crossrefs

For constant term in expansion of Product_{k=1..n} (x^k + 1/x^k)^q for other values of q see A063865, A047653, A124995.

Programs

  • PARI
    a(n) = polcoef(prod(k=1, n, (x^k + 1/x^k)^4), 0); \\ Michel Marcus, Jan 07 2021

Formula

a(n) ~ sqrt(3) * 16^n / (sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 07 2021

A258797 a(n) = [x^n] Product_{k=1..n} (1+x^k)^2 / x^k.

Original entry on oeis.org

1, 1, 2, 6, 16, 51, 166, 554, 1896, 6595, 23212, 82582, 296393, 1071738, 3900696, 14278074, 52526972, 194108087, 720197524, 2681854490, 10019539112, 37545876368, 141080872362, 531457445806, 2006678785762, 7593123695669, 28789152013570, 109356019134584
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 10 2015

Keywords

Comments

a(n) is half the number of subsets of {-n..n} whose sum is n. - Ilya Gutkovskiy, Jul 09 2025

Links

Crossrefs

Cf. A022567, A047653, A258798, A258799.

Programs

  • Maple
    b:= proc(n, s) option remember; `if`(n*(n+1)/2 b(n$2):
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 14 2025
  • Mathematica
    Table[SeriesCoefficient[Product[(1+x^k)^2/x^k, {k, 1, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[Product[1+x^k, {k, 1, n}]^2, {x, 0, n*(n+3)/2}], {n, 0, 30}]

Formula

a(n) ~ sqrt(3) * 4^n / (sqrt(Pi) * n^(3/2)).

A350282 a(n) is the constant term in the expansion of Product_{k=1..n} (x^k + 1/x^k)^n.

Original entry on oeis.org

1, 0, 4, 62, 4658, 0, 2319512420, 14225426190522, 361926393013029354, 0, 16231015449888734994721650504, 28330316118212024049511095643949434, 200866780133770636272812495083578779133456, 0
Offset: 0

Views

Author

Seiichi Manyama, Dec 23 2021

Keywords

Comments

a(n) is the coefficient of x^(n^2 * (n+1)/2) in Product_{k=0..n} (1 + x^(2*k))^n.

Links

Crossrefs

Cf. A063865, A047653, A124995, A124996.
Cf. A350283.

Programs

  • PARI
    a(n) = polcoef(prod(k=1, n, x^k+1/x^k)^n, 0);
  • PARI
    a(n) = polcoef(prod(k=1, n, 1+x^(2*k))^n, n^2*(n+1)/2);

Formula

a(4*n+1) = 0.

A350495 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^2) + 1/x^(k^2))^2.

Original entry on oeis.org

1, 2, 4, 8, 16, 40, 88, 222, 570, 1564, 4516, 13874, 41866, 137432, 442964, 1492610, 4998674, 17204844, 59175316, 207299554, 727137516, 2582078416, 9179001124, 32943918428, 118453240846, 428937325964, 1556421977612, 5676923326262, 20754245720206
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 28 2022

Keywords

Links

Crossrefs

Cf. A000980, A047653, A158092, A350249, A350881.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1,
      expand((x^(n^2)+1/x^(n^2))^2*b(n-1)))
    end:
a:= n-> coeff(b(n),x,0):
seq(a(n), n=0..28);  # Alois P. Heinz, Jan 28 2022
  • Mathematica
    Table[Coefficient[Product[(x^(k^2) + 1/x^(k^2))^2, {k, 1, n}], x, 0], {n, 0, 30}] (* Vaclav Kotesovec, Feb 05 2022 *)

Formula

Conjecture: a(n) ~ sqrt(5) * 4^n / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Feb 05 2022

A369710 Maximal coefficient of (1 - x)^2 * (1 - x^2)^2 * (1 - x^3)^2 * ... * (1 - x^n)^2.

Original entry on oeis.org

1, 1, 4, 3, 10, 6, 20, 12, 34, 24, 64, 52, 116, 103, 208, 223, 410, 470, 808, 992, 1620, 2120, 3352, 4494, 6980, 9584, 14680, 20400, 31128, 43774, 66288, 93968, 141654, 201766, 303716, 433746, 652612, 936334, 1404920, 2021344, 3029564, 4364300, 6541872, 9437054
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2024

Keywords

Crossrefs

Cf. A002107, A047653, A086376.

Programs

  • Mathematica
    Table[Max[CoefficientList[Product[(1 - x^k)^2, {k, 1, n}], x]], {n, 0, 43}]
  • PARI
    a(n) = vecmax(Vec(prod(k=1, n, (1-x^k)^2))); \\ Michel Marcus, Jan 30 2024

A350881 a(n) is the constant term in expansion of Product_{k=1..n} (x^prime(k) + 1/x^prime(k))^2.

Original entry on oeis.org

1, 2, 4, 10, 24, 50, 140, 368, 1152, 3682, 11784, 39902, 134612, 463066, 1635092, 5818384, 20684072, 73693068, 266943648, 967762792, 3533666568, 13036452946, 48102671884, 178315730764, 661567489568, 2450447537226, 9123572154720, 34201574126260
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 20 2022

Keywords

Crossrefs

Cf. A000980, A022894, A022896, A047653.

Programs

  • Maple
    p:= proc(n) option remember; `if`(n=0, 1,
      p(n-1)*(x^ithprime(n)+1/x^ithprime(n))^2)
    end:
a:= n-> coeff(p(n), x, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 26 2022
  • Mathematica
    p[n_] := p[n] = If[n == 0, 1, p[n - 1]*(x^Prime[n] + 1/x^Prime[n])^2];
a[n_] := Coefficient[p[n], x, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 01 2022, after Alois P. Heinz *)
  • PARI
    a(n) = polcoef (prod(k=1, n, (x^prime(k) + 1/x^prime(k))^2), 0); \\ Michel Marcus, Jan 21 2022

A369712 Maximal coefficient of (1 + x) * (1 + x^2)^2 * (1 + x^3)^3 * ... * (1 + x^n)^n.

Original entry on oeis.org

1, 1, 2, 9, 79, 1702, 78353, 7559080, 1509040932, 619097417818, 519429629728698, 887531129680197018, 3078434842626707386602, 21627792113204714623569767, 307257554772242590850211062866, 8813577747274880345454470354985336, 509819403352972623999938010230619997952
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2024

Keywords

Crossrefs

Cf. A025591, A026007, A047653.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1, expand(b(n-1)*(1+x^n)^n)) end:
a:= n-> max(coeffs(b(n))):
seq(a(n), n=0..16);  # Alois P. Heinz, Jan 29 2024
  • Mathematica
    Table[Max[CoefficientList[Product[(1 + x^k)^k, {k, 1, n}], x]], {n, 0, 16}]
  • PARI
    a(n) = vecmax(Vec(prod(k=1, n, (1+x^k)^k))); \\ Michel Marcus, Jan 30 2024

A369725 Maximal coefficient of ( (1 + x) * (1 + x^2) * (1 + x^3) * ... * (1 + x^n) )^n.

Original entry on oeis.org

1, 1, 4, 62, 4658, 1585430, 2319512420, 14225426190522, 361926393013029354, 37883831957216781279561, 16231015449888734994721650504, 28330316118212024049511095643949434, 200866780133770636272812495083578779133456, 5771133366532656054669819186294860881172794669798
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2024

Keywords

Crossrefs

Cf. A025591, A047653, A270913, A369709.

Programs

  • Maple
    a:= n-> max(coeffs(expand(mul(1+x^k, k=1..n)^n))):
seq(a(n), n=0..14);  # Alois P. Heinz, Jan 30 2024
  • Mathematica
    Table[Max[CoefficientList[Product[(1 + x^k)^n, {k, 1, n}], x]], {n, 0, 13}]
  • PARI
    a(n) = vecmax(Vec(prod(k=1, n, (1+x^k))^n)); \\ Michel Marcus, Jan 30 2024

A380499 Absolute value of the minimum coefficient of (1 - x)^2 * (1 - x^2)^2 * (1 - x^3)^2 * ... * (1 - x^n)^2.

Original entry on oeis.org

1, 2, 2, 6, 4, 12, 8, 24, 19, 44, 36, 78, 74, 148, 156, 286, 322, 556, 682, 1120, 1448, 2308, 3072, 4784, 6538, 10064, 14001, 21296, 29928, 45276, 64032, 96712, 137520, 207156, 296236, 444748, 637812, 956884, 1373622, 2062080, 2968872, 4450120, 6422472, 9616202, 13894990, 20802836
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 25 2025

Keywords

Crossrefs

Cf. A002107, A047653, A086394, A133871, A369710.

Programs

  • Maple
    p:= proc(n) option remember;
     `if`(n=0, 1, expand(p(n-1)*(1-x^n)^2))
    end:
a:= n-> abs(min(coeffs(p(n)))):
seq(a(n), n=0..45);  # Alois P. Heinz, Jan 25 2025
  • Mathematica
    Table[Min[CoefficientList[Product[(1 - x^k)^2, {k, 1, n}], x]], {n, 0, 45}] // Abs
  • PARI
    a(n) = abs(vecmin(Vec(prod(k=1, n, (1-x^k)^2)))); \\ Michel Marcus, Jan 25 2025
Previous Showing 11-20 of 24 results. Next

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

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