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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A359389 Maximal coefficient of Product_{k=1..n} (1 + 2*x^k).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 72, 176, 384, 976, 2496, 6560, 17152, 45952, 123520, 336640, 920832, 2526976, 6979584, 19379712, 53966336, 150892544, 423132160, 1190260736, 3356964864, 9491228672, 26889519104, 76351971328, 217229369344, 619159953408, 1767696515072, 5054679908352
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 29 2022

Keywords

Links

Crossrefs

Cf. A025591, A032302, A160235.

Programs

  • Mathematica
    Table[Max[CoefficientList[Product[1 + 2*x^k, {k, 1, n}], x]], {n, 0, 40}]
p = 1; Join[{1}, Table[p = Expand[p*(1 + 2*x^n)]; Max[CoefficientList[p, x]], {n, 1, 40}]]
  • PARI
    a(n) = vecmax(Vec(prod(k=1, n, 1 + 2*x^k))); \\ Michel Marcus, Dec 29 2022

Formula

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

A367087 Number of solutions to +- 1 +- 3 +- 5 +- 7 +- ... +- (2*n-1) = 0 or 1.

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 2, 5, 8, 13, 20, 38, 68, 118, 206, 380, 692, 1262, 2306, 4277, 7930, 14745, 27492, 51541, 96792, 182182, 343670, 650095, 1231932, 2338706, 4447510, 8472697, 16164914, 30884150, 59086618, 113189168, 217091832, 416839177, 801247614, 1541726967, 2969432270
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 26 2024

Keywords

Crossrefs

Cf. A005408, A025591, A156700, A292476.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n>i^2, 0,
      `if`(i=0, 1, b(n+2*i-1, i-1)+b(abs(n-2*i+1), i-1)))
    end:
a:=n-> b(irem(n, 2), n):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 26 2024
  • Mathematica
    b[n_, i_] := b[n, i] = If[n > i^2, 0,
   If[i == 0, 1, b[n+2*i-1, i-1] + b[Abs[n-2*i+1], i-1]]];
a[n_] := b[Mod[n, 2], n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 03 2025, after Alois P. Heinz *)

A369705 Maximal coefficient of (1 + x) * (1 - x^2) * (1 + x^3) * ... * (1 - (-x)^n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 3, 3, 4, 6, 5, 6, 7, 7, 8, 10, 11, 16, 16, 19, 21, 23, 28, 34, 41, 50, 56, 68, 80, 91, 110, 135, 158, 196, 225, 269, 320, 376, 447, 544, 644, 786, 921, 1111, 1321, 1573, 1882, 2274, 2711, 3280, 3895, 4694, 5591, 6718
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2024

Keywords

Crossrefs

Cf. A025591, A039828, A086376, A121373.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1, expand(b(n-1)*(1-(-x)^n))) end:
a:= n-> max(coeffs(b(n))):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 29 2024
  • Mathematica
    Table[Max[CoefficientList[Product[(1 - (-x)^k), {k, 1, n}], x]], {n, 0, 60}]
  • PARI
    a(n) = vecmax(Vec(prod(k=1, n, (1-(-x)^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

A369765 Maximal coefficient of (1 + x) * (1 + x^2) * (1 + x^3) * (1 + x^5) * ... * (1 + x^prime(n-1)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 6, 7, 13, 19, 32, 53, 90, 156, 277, 494, 878, 1566, 2836, 5146, 9401, 17358, 32042, 59434, 110292, 204332, 380548, 713601, 1342448, 2538012, 4808578, 9043605, 17070234, 32268611, 61271738, 116123939, 220993892, 421000142, 802844420, 1534312896
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 31 2024

Keywords

Crossrefs

Cf. A008578, A025591, A036497, A350457, A350514, A369708.

Programs

  • Mathematica
    Table[Max[CoefficientList[Product[(1 + x^If[k == 1, 1, Prime[k - 1]]), {k, 1, n}], x]], {n, 0, 40}]
  • Python
    from collections import Counter
from sympy import prime
def A369765(n):
    c = {0:1,1:1}
    for k in range(1,n):
        p, d = prime(k), Counter(c)
        for j in c:
            d[j+p] += c[j]
        c = d
    return max(c.values()) # Chai Wah Wu, Feb 01 2024

A369766 Maximal coefficient of Product_{i=1..n} Sum_{j=0..i} x^(i*j).

Original entry on oeis.org

1, 1, 1, 2, 6, 24, 115, 662, 4456, 34323, 298220, 2885156, 30760556, 358379076, 4530375092, 61762729722, 903311893770, 14108704577103, 234387946711329, 4127027097703638, 76774080851679152, 1504640319524566870, 30986929089570280955, 669023741837953551188
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 31 2024

Keywords

Crossrefs

Cf. A000140, A025591, A052335.

Programs

  • Maple
    a:= n-> max(coeffs(expand(mul(add(x^(i*j), j=0..i), i=1..n)))):
seq(a(n), n=0..23);  # Alois P. Heinz, Jan 31 2024
  • Mathematica
    Table[Max[CoefficientList[Product[Sum[x^(i j), {j, 0, i}], {i, 1, n}], x]], {n, 0, 23}]
  • PARI
    a(n) = vecmax(Vec(prod(i=1, n, sum(j=0, i, x^(i*j))))); \\ Michel Marcus, Jan 31 2024
  • Python
    from collections import Counter
def A369766(n):
    c = {0:1,1:1}
    for i in range(2,n+1):
        d = Counter()
        for k in c:
            for j in range(0,i*i+1,i):
                d[j+k] += c[k]
        c = d
    return max(c.values()) # Chai Wah Wu, Jan 31 2024

A369767 Maximal coefficient of Product_{i=1..n} Sum_{j=0..n} x^(i*j).

Original entry on oeis.org

1, 1, 2, 6, 31, 231, 2347, 29638, 449693, 7976253, 162204059, 3722558272, 95221978299, 2687309507102, 82967647793153, 2782190523572392, 100715040802229833, 3914979746952224303, 162662679830709439637, 7194483479557973730982, 337519906320930133470189
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 31 2024

Keywords

Crossrefs

Cf. A000041, A025591, A077047.

Programs

  • Maple
    a:= n-> max(coeffs(expand(mul(add(x^(i*j), j=0..n), i=1..n)))):
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 31 2024
  • Mathematica
    Table[Max[CoefficientList[Product[Sum[x^(i j), {j, 0, n}], {i, 1, n}], x]], {n, 0, 20}]
  • PARI
    a(n) = vecmax(Vec(prod(i=1, n, sum(j=0, n, x^(i*j))))); \\ Michel Marcus, Jan 31 2024
  • Python
    from collections import Counter
def A369767(n):
    c = {j:1 for j in range(n+1)}
    for i in range(2,n+1):
        d = Counter()
        for k in c:
            for j in range(0,i*n+1,i):
                d[j+k] += c[k]
        c = d
    return max(c.values()) # Chai Wah Wu, Jan 31 2024

A068202 Next-to-middle coefficient in expansion of Product_{k=1..n} (1 + x^k).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 23, 39, 69, 123, 219, 396, 719, 1313, 2406, 4435, 8215, 15260, 28431, 53167, 99774, 187615, 353500, 667874, 1264854, 2399207, 4557831, 8675836, 16544234, 31587644, 60382450, 115601178, 221625505, 425313967
Offset: 1

Views

Author

N. J. A. Sloane, Mar 24 2002

Keywords

Crossrefs

Cf. A025591 for middle coefficient.

Programs

  • PARI
    a(n) = polcoeff(prod(k=1, n, 1+x^k), n*(n+1)\4 - 1); \\ Michel Marcus, Jan 04 2021

A367088 Number of solutions to +- 1 +- 2 +- 3 +- 5 +- 7 +- ... +- prime(n-1) = 0 or 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 6, 7, 12, 19, 32, 53, 90, 156, 276, 493, 878, 1566, 2834, 5146, 9396, 17358, 32042, 59434, 110292, 204332, 380548, 713601, 1342448, 2538012, 4808578, 9043605, 17070234, 32268611, 61271738, 116123939, 220993892, 421000142, 802844420, 1534312896
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 26 2024

Keywords

Crossrefs

Cf. A008578, A025591, A022894, A022895, A113040, A350404.

A369706 Maximal coefficient of (1 + x^2) * (1 + x^3) * (1 + x^4) * ... * (1 + x^n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 7, 12, 20, 35, 62, 112, 199, 361, 657, 1206, 2221, 4110, 7636, 14234, 26618, 49910, 93846, 176906, 334184, 632602, 1199892, 2280164, 4340064, 8273610, 15796439, 30202620, 57820648, 110826888, 212681976, 408610024, 785833480, 1512776590, 2915017360
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2024

Keywords

Crossrefs

Cf. A025147, A025591.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n<2, 1, expand(b(n-1)*(1+x^n))) end:
a:= n-> max(coeffs(b(n))):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 29 2024
  • Mathematica
    Table[Max[CoefficientList[Product[(1 + x^k), {k, 2, n}], x]], {n, 0, 40}]
  • PARI
    a(n) = vecmax(Vec(prod(i=2, n, 1+x^i))); \\ Michel Marcus, Jan 29 2024

Formula

a(n) ~ A025591(n)/2. - Vaclav Kotesovec, Jan 29 2024
Previous Showing 41-50 of 51 results. Next

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

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