A001401 -id:A001401 - cp's OEIS frontend

A347587 Number of partitions of n into at most 5 distinct parts.

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 75, 88, 102, 119, 137, 158, 181, 207, 235, 268, 302, 341, 383, 430, 480, 536, 595, 661, 731, 808, 889, 979, 1073, 1176, 1285, 1403, 1527, 1662, 1803, 1956, 2116, 2288, 2468, 2662, 2864, 3080, 3306, 3547

Offset: 0

Ilya Gutkovskiy, Sep 08 2021

Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1).

Cf. A001401, A014591, A065033, A347586, A347588.

nmax = 58; CoefficientList[Series[Sum[x^(k (k + 1)/2)/Product[(1 - x^j), {j, 1, k}], {k, 0, 5}], {x, 0, nmax}], x]
LinearRecurrence[{1, 1, 0, 0, -1, -1, -1, 1, 1, 1, 0, 0, -1, -1, 1}, {1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22}, 59]

G.f.: Sum_{k=0..5} x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

A008766 Expansion of (1+x^5)/((1-x)(1-x^2)(1-x^3)*(1-x^4)).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 29, 36, 45, 54, 65, 77, 91, 106, 123, 141, 162, 184, 208, 234, 263, 293, 326, 361, 399, 439, 482, 527, 576, 627, 681, 738, 799, 862, 929, 999, 1073, 1150, 1231, 1315, 1404, 1496, 1592, 1692, 1797, 1905, 2018, 2135, 2257, 2383, 2514, 2649, 2790, 2935

Offset: 0

N. J. A. Sloane

From Washington Bomfim, Jan 14 2021: (Start)
Let \n,m\ be the number of partitions of n into m non-distinct parts.
For n >= 1, \n,5\ = round((2*n^3-15*n^2+60*n-110*[n mod 2 = 0]-65*[n mod 2])/144).
For n >= 10, \n,5\ = A026811(n) - A026811(n-10).
(End)

Washington Bomfim, Table of n, a(n) for n = 0..9999 (first 1000 terms from G. C. Greubel)
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1,-1,1,-1,2,-1).

Cf. A026811, A001401.

a:=[1,1,2,3,5,7,10,13,18];; for n in [10..60] do a[n]:=2*a[n-1]-a[n-2]+a[n-3]-a[n-4]-a[n-5]+a[n-6]-a[n-7]+2*a[n-8]-a[n-9]; od; a; # G. C. Greubel, Sep 10 2019

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) )); // G. C. Greubel, Sep 10 2019

seq(coeff(series((1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)), x, n+1), x, n), n = 0 .. 60); # G. C. Greubel, Sep 10 2019

CoefficientList[Series[(1+x^5)/(1-x)/(1-x^2)/(1-x^3)/(1-x^4),{x,0,60}],x] (* or *) LinearRecurrence[{2,-1,1,-1,-1,1,-1,2,-1}, {1,1,2,3,5,7,10, 13,18}, 60] (* Harvey P. Dale, Jul 24 2016 *)

Vec((1+x^5)/(1-x)/(1-x^2)/(1-x^3)/(1-x^4) +O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

seq(x) = { a = vector(x+1); my(N = 5);
for(n=0,x, a[n+1]=round((2*N^3-15*N^2+60*N-110*!(N%2)-65*(N%2))/144); N++);a};
seq(60) \\ Washington Bomfim, Jan 14 2021

def A008766_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))).list()
A008766_list(60) # G. C. Greubel, Sep 10 2019

a(n) = round((2*N^3 - 15*N^2 + 60*N - 110*[N mod 2=0] - 65*[N mod 2])/144), where N = n+5. - Washington Bomfim, Jan 14 2021

Terms a(45) onward added by G. C. Greubel, Sep 10 2019

A050910 Number of pure 3-complexes on 8 unlabeled nodes with n multiple 3-simplexes.

Original entry on oeis.org

1, 1, 5, 21, 131, 940, 7902, 69025, 594203, 4856288, 37189863, 265916174, 1778005595, 11154474602, 65921168577, 368463685296, 1955231394323, 9884229508860, 47752849848446, 221109815065563, 983764137502726

Offset: 0

Vladeta Jovovic, Dec 29 1999

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A001401, A014396.

\\ See A304942 for Blocks
a(n)={Blocks(n,8,4)} \\ Andrew Howroyd, Feb 09 2020

A256525 Number of partitions of 3n into at most 5 parts.

Original entry on oeis.org

1, 3, 10, 23, 47, 84, 141, 221, 333, 480, 674, 918, 1226, 1602, 2062, 2611, 3266, 4033, 4932, 5969, 7166, 8529, 10083, 11835, 13811, 16019, 18487, 21224, 24260, 27604, 31289, 35324, 39744, 44559, 49806, 55496, 61667, 68331, 75529, 83273, 91606, 100540

Offset: 0

Colin Barker, Apr 01 2015

			For n=1 the 3 partitions of 1*3 = 3 are [3], [1,2] and [1,1,1].

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-1,-2,2,1,-2,2,0,-2,1).

Cf. A001401, A077043 (3 parts), A256524 (4 parts), A256315 (6 parts).

Table[Length[IntegerPartitions[3n,5]],{n,0,50}] (* Harvey P. Dale, Mar 08 2019 *)

concat(1, vector(40, n, k=0; forpart(p=3*n, k++, , [1,5]); k))

Vec(-(x^8+x^7+4*x^6+5*x^5+5*x^4+5*x^3+4*x^2+x+1) / ((x-1)^5*(x+1)^2*(x^2+1)*(x^4+x^3+x^2+x+1)) + O(x^100))

G.f.: -(x^8+x^7+4*x^6+5*x^5+5*x^4+5*x^3+4*x^2+x+1) / ((x-1)^5*(x+1)^2*(x^2+1)*(x^4+x^3+x^2+x+1)).

a(n) = A001401(3n). - Alois P. Heinz, Apr 01 2015

A256539 Number of partitions of 4n into at most 5 parts.

Original entry on oeis.org

1, 5, 18, 47, 101, 192, 333, 540, 831, 1226, 1747, 2418, 3266, 4319, 5608, 7166, 9027, 11229, 13811, 16814, 20282, 24260, 28796, 33940, 39744, 46262, 53550, 61667, 70673, 80631, 91606, 103664, 116875, 131310, 147042, 164147, 182702, 202787, 224484, 247877

Offset: 0

Colin Barker, Apr 01 2015

			For n=2 the 18 partitions of 2*4 = 8 are [8], [1,7], [2,6], [3,5], [4,4], [1,1,6], [1,2,5], [1,3,4], [2,2,4], [2,3,3], [1,1,1,5], [1,1,2,4], [1,1,3,3], [1,2,2,3], [2,2,2,2], [1,1,1,1,4], [1,1,1,2,3] and [1,1,2,2,2].

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,4,-4,3,-2,3,-3,1).

Cf. A001401, A238340 (4 parts), A256540 (6 parts).

concat(1, vector(40, n, k=0; forpart(p=4*n, k++, , [1,5]); k))

Vec(-(x^7+4*x^6+5*x^5+7*x^4+6*x^3+6*x^2+2*x+1) / ((x-1)^5*(x^2+x+1)*(x^4+x^3+x^2+x+1)) + O(x^100))

G.f.: -(x^7+4*x^6+5*x^5+7*x^4+6*x^3+6*x^2+2*x+1) / ((x-1)^5*(x^2+x+1)*(x^4+x^3+x^2+x+1)).

a(n) = A001401(4n). - Alois P. Heinz, Apr 01 2015

A274322 Number of partitions of n^2 into at most five parts.

Original entry on oeis.org

1, 1, 5, 23, 101, 377, 1226, 3507, 9027, 21224, 46262, 94512, 182702, 336666, 595085, 1014091, 1673243, 2682685, 4192118, 6401314, 9572962, 14047457, 20260601, 28763703, 40247228, 55567352, 75776769, 102158957, 136267461, 179969238, 235493851, 305487369

Offset: 0

Colin Barker, Jun 20 2016

nonn

Colin Barker, Table of n, a(n) for n = 0..1000

A subsequence of A001401.

\\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)).
b(n) = round(((n+5)^4+10*((n+5)^3+(n+5)^2)-75*(n+5)-45*(n+5)*(-1)^(n+5))/2880)
vector(40, n, n--; b(n^2))

Coefficient of x^(n^2) in 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)).
a(n) = A001401(n^2).
Empirical g.f.: (1 -3*x +4*x^2 +13*x^3 +21*x^4 +63*x^5 +138*x^6 +204*x^7 +257*x^8 +280*x^9 +267*x^10 +201*x^11 +128*x^12 +67*x^13 +31*x^14 +6*x^15 +x^16 +x^17) / ((1 -x)^9*(1 +x)^3*(1 +x +x^2)*(1 +x +x^2 +x^3 +x^4)).

A307449 Irregular triangle read by rows: T(n, k) gives the coefficients of the Girard-Waring formula for the sum of n-th power of five indeterminates in terms of their elementary symmetric functions (reverse Abramowitz-Stegun order of partitions).

Original entry on oeis.org

1, 1, -2, 1, -3, 3, 1, -4, 2, 4, -4, 1, -5, 5, 5, -5, -5, 5, 1, -6, 9, 6, -2, -12, -6, 3, 6, 6, 1, -7, 14, 7, -7, -21, -7, 7, 7, 14, 7, -7, -7, 1, -8, 20, 8, -16, -32, -8, 2, 24, 12, 24, 8, -8, -8, -16, -16, 4, 8, 1, -9, 27, 9, -30, -45, -9, 9, 54, 18, 36, 9, -9, -27, -27, -27, -27, 3, 18, 9, 9, 18, -9, 1, -10, 35, 10, -50, -60, -10, 25, 100, 25, 50, 10, -2, -40, -60, -60, -40, -40, 15, 10, 10, 60, 30, 15, 30, -10, -10, -20, -20, 5

Offset: 1

Wolfdieter Lang, May 14 2019

The length of row n is A001401(n), n >= 1.
The Girard-Waring formula for the power sum p(5,n) = Sum_{j=1..5} (x_j)^n in terms of the elementary symmetric functions e_j(x_1, x_2, x_3, x_4), for j = 1, 2 ,..., 5 is given in the W. Lang reference, Theorem 1, in an explicitly nested four sums version. See also the summary link, for N = 5 (there sigma_j^{(N)} -> e_j here).
In this array the partitions of n, with all partitions with a part >= 6 omitted, are used. Here the partitions appear in the reverse Abramowitz-Stegun order. See row n of the array of Waring numbers A115131, read backwards, with the entries corresponding to these omitted partitions.

			The irregular triangle T(n, k) begins:
n\k 1   2  3  4   5   6  7 8  9 10 11 12 13  14  15  16  17 18 19 20 21 22 23
-----------------------------------------------------------------------------
1:  1
2:  1  -2
3:  1  -3  3
4:  1  -4  2  4  -4
5:  1  -5  5  5  -5  -5  5
6:  1  -6  9  6  -2 -12 -6 3  6  6
7:  1  -7 14  7  -7 -21 -7 7  7 14  7 -7 -7
8:  1  -8 20  8 -16 -32 -8 2 24 12 24  8 -8  -8 -16 -16   4  8
9:  1  -9 27  9 -30 -45 -9 9 54 18 36  9 -9 -27 -27 -27 -27  3 18  9  9 18 -9
.
.
.
n = 10: 1 -10 35 10 -50 -60 -10 25 100 25 50 10 -2 -40 -60 -60 -40 -40 15 10 10 60 30 15 30 -10 -10 -20 -20 5.
...
------------------------------------------------------------------------------
Row n = 6: x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 =  1*e_1^6  - 6*e_1^4*e_2 + 9*e_1^2*e_2^2 + 6*e_1^3*e_3 - 2*e_2^3 - 12*e_1*e_2*e_3 - 6*e_1^2*e_4 + 3*e_3^2 + 6*e_2*e_4 + 6*e_1*e_5,  with e_1 = Sum_{j=1..5} x_j, e_2 = x1*(x_2 + x_3 + x_4 + x_5) + x_2*(x_3 + x_4 + x_5) + x_3*(x_4 + x_5) + x_4*x_5, e_3 = x_1*x_2*x_3 + x_1*x_2*x_4 +  x_1*x_2*x_5 +  x_2*x_3*x_4 + x_2*x_3*x_5 + x_2*x_4*x_5 + x_3*x_4*x_5, e_4 =  x_1*x_2*x_3*x_4 + x_1*x_2*x_3*x_5 + x_1*x_2*x_4*x_5 + x_1*x_3*x_4*x_5 + x_2*x_3*x_4*x_5, e_5 = Product_{i=1..5} x_j.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]
Wolfdieter Lang, On sums of powers of zeros of polynomials, J. Comp. Appl. Math. 89 (1998) 237-256; Theorem 1.
Wolfdieter Lang, Nested sum version of the Girard-Waring formula (a summary)

Cf. A001401, A115131, A132460 (N=2), A325477 (N=3), A324602 (N=4).

T(n, k) is the k-th coefficient of the Waring number partition array A115131(n, m) (k there is replaced here by m), read backwards, omitting all partitions which have a part >= 6.

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A347587 Number of partitions of n into at most 5 distinct parts.

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A008766 Expansion of (1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).

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A050910 Number of pure 3-complexes on 8 unlabeled nodes with n multiple 3-simplexes.

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A256525 Number of partitions of 3n into at most 5 parts.

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A256539 Number of partitions of 4n into at most 5 parts.

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A274322 Number of partitions of n^2 into at most five parts.

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A307449 Irregular triangle read by rows: T(n, k) gives the coefficients of the Girard-Waring formula for the sum of n-th power of five indeterminates in terms of their elementary symmetric functions (reverse Abramowitz-Stegun order of partitions).

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A008766 Expansion of (1+x^5)/((1-x)(1-x^2)(1-x^3)*(1-x^4)).