A086781 -id:A086781 - cp's OEIS frontend

A086376 Maximal coefficient of the polynomial (1-x)(1-x^2)...*(1-x^n).

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 4, 3, 3, 4, 6, 5, 6, 7, 8, 8, 10, 11, 16, 16, 18, 21, 28, 29, 34, 41, 50, 56, 66, 80, 100, 114, 131, 158, 196, 225, 263, 320, 388, 455, 532, 644, 786, 921, 1083, 1321, 1600, 1891, 2218, 2711, 3280, 3895, 4588, 5591, 6780, 8051, 9519, 11624

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 07 2003

Alois P. Heinz, Table of n, a(n) for n = 0..1000
S. R. Finch, Signum equations and extremal coefficients.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Volume 4, No.1, December 2008. pp. 176-187. [From _Johannes W. Meijer_, Jun 21 2010]
E. M. Wright, A closer estimate for a restricted partition function, Q. J. Math. 15 (1964) 283-287.

Cf. A025591, A086781, A160089.

A086376 := proc(n)
        g := expand(mul( 1-x^k,k=1..n) );
        convert(PolynomialTools[CoefficientVector](g, x), list):
        max(%);
end proc:
seq(A086376(n),n=0..60) ; # R. J. Mathar, Jun 01 2011

b[0] = 1; b[n_] := b[n] = b[n-1]*(1-x^n) // Expand;
a[n_] := CoefficientList[b[n], x] // Max;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 13 2017 *)

a(n)=vecmax(Vec(prod(k=1,n,1-x^k)));
vector(100,n,a(n-1)) \\ Joerg Arndt, Jan 31 2024

More terms from Sascha Kurz, Sep 22 2003

a(0)=1 prepended by Alois P. Heinz, Apr 12 2017

A086795 a(n) is the number of terms in the expansion of (x-2y)(x^2-2y^2)(x^3-2y^3)...(x^n-2y^n).

Original entry on oeis.org

2, 4, 7, 11, 13, 21, 28, 35, 42, 53, 65, 74, 87, 102, 117, 133, 150, 168, 188, 207, 228, 251, 273, 298, 323, 348, 375, 404, 433, 463, 494, 526, 559, 592, 627, 664, 701, 739, 778, 817, 858, 901, 944, 988, 1032, 1079, 1126, 1174, 1222, 1273

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 05 2003

nonn

Cf. A086781.

A086795 := proc(n)
    mul( x^i-2*y^i,i=1..n) ;
    expand(%) ;
    nops([coeffs(%)]) ;
end proc: # R. J. Mathar, Sep 15 2012

A225549 a(n) is the number of terms in the expansion of (x-y)(x^4-y^4)(x^9-y^9)...(x^(n^2)-y^(n^2)).

Original entry on oeis.org

2, 4, 8, 16, 24, 40, 68, 103, 162, 236, 344, 453, 612, 790, 994, 1229, 1432, 1782, 2134, 2517, 2968, 3460, 3974, 4543, 5160, 5822, 6546, 7347, 8184, 9080, 10058, 11075, 12166, 13316, 14536, 15837, 17202, 18654, 20156, 21765, 23450, 25212, 27074, 29001, 31032, 33158, 35370, 37679, 40070, 42578

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com) and Robert G. Wilson v, May 10 2013

nonn

In the definition one can take y=1. Thus the sequence becomes the number of terms in the polynomial of the product{k=1..n} (1-x^(k^2)).

Cf. A000124, A086781, A086795, A086796, A086817.

a[n_] := Length@ ExpandAll@ Product[(1 - x^(k^2)), {k, n}]; Array[f, 40]

a(n)=my(P=prod(k=1,n,1-'x^k^2)); sum(i=0, poldegree(P), polcoeff(P,i)!=0) \\ Charles R Greathouse IV, May 10 2013

A221991 a(n) is the number of terms in the expansion of (x-y)(x^2-y^2)(x^3-y^3)(x^5-y^5)...(x^p_i-y^p_i), where p_i is the i-th prime.

Original entry on oeis.org

2, 4, 6, 8, 10, 20, 22, 36, 42, 66, 90, 110, 142, 184, 232, 284, 342, 400, 458, 532, 604, 678, 756, 838, 928, 1026, 1126, 1230, 1336, 1446, 1558, 1686, 1816, 1954, 2092, 2242, 2392, 2550, 2712, 2880, 3052, 3232, 3412, 3604, 3796, 3994, 4192, 4404, 4626, 4854, 5082

Offset: 1

Robert G. Wilson v, May 12 2013

nonn

In the definition one can take y=1. Thus the sequence becomes the number of terms in the polynomial of the product{k=0..n} (1-x^p_i), where p_i is the i-th prime and p_0 = 1.

Offset is 1 to keep it parallel to other like sequences.

Cf. A000124, A086781, A086795, A086796, A086817, A225549.

f[n_] := Length@ ExpandAll[(1 - x) Product[(1 - x^Prime[k]), {k, n}]]; Array[f, 51, 0]

A301705 a(n) is the number of zero coefficients of the polynomial (x-1)(x^2-1)...*(x^n-1) below the leading coefficient.

Original entry on oeis.org

0, 0, 1, 4, 4, 8, 11, 12, 14, 20, 25, 26, 24, 42, 37, 40, 46, 46, 45, 50, 62, 62, 69, 72, 80, 78, 79, 74, 88, 94, 97, 102, 94, 104, 105, 106, 102, 116, 137, 130, 126, 132, 121, 122, 134, 152, 155, 160, 164, 156, 143, 156, 170, 172, 167, 178, 186, 194, 185, 168, 174, 176, 183, 182, 192, 194, 205, 196, 200, 188

Offset: 1

Ovidiu Bagdasar, Mar 25 2018

			Denote P_n(x) = (x-1)...(x^n-1).
P_1(x) = x-1, hence a(1)=0.
P_2(x) = (x-1)*(x^2-1) = x^3-x^2-x+1, hence a(2)=0;
P_3(x) = (x-1)*(x^2-1)*(x^3-1) = x^6-x^5-x^4+x^2+x-1, hence a(3)=1;
P_4(x) = (x-1)*(x^2-1)*(x^3-1)*(x^4-1) = x^10 - x^9 - x^8+2x^5-x^2-x+1, hence a(4)=4.

Michael De Vlieger, Table of n, a(n) for n = 1..300
Dorin Andrica and Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.

Cf. A086376, A086781, A231599.

a:= n-> add(`if`(i=0, 1, 0), i=[(p-> seq(coeff(p, x, i),
         i=0..degree(p)))(expand(mul(x^i-1, i=1..n)))]):
seq(a(n), n=1..70);  # Alois P. Heinz, Mar 29 2019

Rest@ Array[Count[CoefficientList[Times @@ Array[x^# - 1 &, # - 1], x], ?(# == 0 &)] &, 71] (* _Michael De Vlieger, Mar 29 2019 *)

a(n) = #select(x->(x==0), Vec((prod(k=1, n, (x^k-1))))); \\ Michel Marcus, Apr 02 2018

a(n) = 1+n(n+1)/2-A086781(n).

A222028 a(n) is the number of terms in the expansion of (x-y)(x^3-y^3)(x^6-y^6)(x^10-y^10)...(x^T_i-y^T_i), where T_i is the i-th triangular number.

Original entry on oeis.org

2, 4, 8, 15, 28, 41, 66, 92, 132, 175, 232, 287, 360, 475, 570, 727, 852, 1009, 1220, 1397, 1646, 1891, 2154, 2441, 2772, 3121, 3508, 3891, 4334, 4791, 5282, 5797, 6376, 6983, 7618, 8285, 8984, 9713, 10500, 11319, 12182, 13093, 14028, 15023, 16064, 17157, 18276, 19447, 20680, 21953

Offset: 1

Robert G. Wilson v, May 12 2013

nonn

In the definition one can take y=1. Thus the sequence becomes the number of terms in the polynomial of the product{k=0..n} (1-x^T_i), where G_i is the i-th triangular number.

Cf. A000124, A086781, A086795, A086796, A086817, A221991, A225549.

f[n_] := Length@ ExpandAll@ Product[1 - x^(k (k + 1)/2), {k, n}]; Array[f, 50]

cp's OEIS Frontend

A086376 Maximal coefficient of the polynomial (1-x)*(1-x^2)*...*(1-x^n).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A086795 a(n) is the number of terms in the expansion of (x-2y)*(x^2-2y^2)*(x^3-2y^3)*...*(x^n-2y^n).

Views

Author

Keywords

Crossrefs

Programs

Maple

A225549 a(n) is the number of terms in the expansion of (x-y)*(x^4-y^4)*(x^9-y^9)*...*(x^(n^2)-y^(n^2)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

A221991 a(n) is the number of terms in the expansion of (x-y)(x^2-y^2)*(x^3-y^3)*(x^5-y^5)*...*(x^p_i-y^p_i), where p_i is the i-th prime.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A301705 a(n) is the number of zero coefficients of the polynomial (x-1)*(x^2-1)*...*(x^n-1) below the leading coefficient.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A222028 a(n) is the number of terms in the expansion of (x-y)(x^3-y^3)*(x^6-y^6)*(x^10-y^10)*...*(x^T_i-y^T_i), where T_i is the i-th triangular number.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A086376 Maximal coefficient of the polynomial (1-x)(1-x^2)...*(1-x^n).

A086795 a(n) is the number of terms in the expansion of (x-2y)(x^2-2y^2)(x^3-2y^3)...(x^n-2y^n).

A225549 a(n) is the number of terms in the expansion of (x-y)(x^4-y^4)(x^9-y^9)...(x^(n^2)-y^(n^2)).

A221991 a(n) is the number of terms in the expansion of (x-y)(x^2-y^2)(x^3-y^3)(x^5-y^5)...(x^p_i-y^p_i), where p_i is the i-th prime.

A301705 a(n) is the number of zero coefficients of the polynomial (x-1)(x^2-1)...*(x^n-1) below the leading coefficient.

A222028 a(n) is the number of terms in the expansion of (x-y)(x^3-y^3)(x^6-y^6)(x^10-y^10)...(x^T_i-y^T_i), where T_i is the i-th triangular number.