A000332 -id:A000332 - cp's OEIS frontend

A000391 Euler transform of A000332.

1, 6, 21, 71, 216, 672, 1982, 5817, 16582, 46633, 128704, 350665, 941715, 2499640, 6557378, 17024095, 43756166, 111433472, 281303882, 704320180, 1749727370, 4314842893, 10565857064, 25700414815, 62115621317, 149214574760, 356354881511, 846292135184

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..500
A. O. L. Atkin, P. Bratley, I. G. McDonald, and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
A. O. L. Atkin, P. Bratley, I. G. McDonald, and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]
Srivatsan Balakrishnan, Suresh Govindarajan, and Naveen S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011, p.21.
N. J. A. Sloane, Transforms

Cf. A000041, A000219, A000294, A000335, A000417, A000428, A255965.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> binomial(n+3,4)): seq(a(n), n=1..30); # Alois P. Heinz, Sep 08 2008

nn = 50; b = Table[Binomial[n, 4], {n, 4, nn + 4}]; Rest[CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x]] (* T. D. Noe, Jun 21 2012 *)
nmax=50; Rest[CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)*(k+2)*(k+3)/24),{k,1,nmax}],{x,0,nmax}],x]] (* Vaclav Kotesovec, Mar 11 2015 *)

a(n)=if(n<0, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)^5/k, x*O(x^n))), n)) /* Joerg Arndt, Apr 16 2010 */

a(n) ~ Pi^(3/160) / (2 * 3^(243/320) * 7^(83/960) * n^(563/960)) * exp(Zeta'(-1)/4 - 143 * Zeta(3) / (240 * Pi^2) + 53461 * Zeta(5) / (3200 * Pi^4) + 107163 * Zeta(3) * Zeta(5)^2 / (2*Pi^12) - 24754653 * Zeta(5)^3 / (10*Pi^14) + 413420708484 * Zeta(5)^5 / (5*Pi^24) + Zeta'(-3)/4 + (-847 * 7^(1/6) * Pi / (19200 * sqrt(3)) - 189 * sqrt(3) * 7^(1/6) * Zeta(3) * Zeta(5) / (2*Pi^7) + 305613 * sqrt(3) * 7^(1/6) * Zeta(5)^2 / (80*Pi^9) - 614365479 * sqrt(3) * 7^(1/6) * Zeta(5)^4 / (4*Pi^19)) * n^(1/6) + (3 * 7^(1/3) * Zeta(3) / (4*Pi^2) - 693 * 7^(1/3) * Zeta(5) / (40*Pi^4) + 857304 * 7^(1/3) * Zeta(5)^3 / Pi^14) * n^(1/3) + (11 * sqrt(7/3) * Pi / 120 - 1701 * sqrt(21) * Zeta(5)^2 / Pi^9) * sqrt(n) + 27 * 7^(2/3) * Zeta(5) / (2*Pi^4) * n^(2/3) + 2*sqrt(3)*Pi / (5*7^(1/6)) * n^(5/6)). - Vaclav Kotesovec, Mar 12 2015

A104392 Sums of 2 distinct positive pentatope numbers (A000332).

Original entry on oeis.org

6, 16, 20, 36, 40, 50, 71, 75, 85, 105, 127, 131, 141, 161, 196, 211, 215, 225, 245, 280, 331, 335, 336, 345, 365, 400, 456, 496, 500, 510, 530, 540, 565, 621, 705, 716, 720, 730, 750, 785, 825, 841, 925, 1002, 1006, 1016, 1036, 1045, 1071, 1127

Offset: 0

Jonathan Vos Post, Mar 05 2005

Pentatope number Ptop(n) = binomial(n+3,4) = n*(n+1)*(n+2)*(n+3)/24. Hyun Kwang Kim asserts that every positive integer can be represented as the sum of no more than 8 pentatope numbers; but in this sequence we are only concerned with sums of nonzero distinct pentatope numbers.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Pentatope Number.

Cf. A000332, A100009, A102856.

nn=15; Select[Union[Total/@Subsets[Binomial[Range[4,nn],4],{2}]], #Harvey P. Dale, Mar 13 2011 *)

a(n) = Ptop(i) + Ptop(j) for some positive i=/=j and Ptop(n) = binomial(n+3,4).

Extended by Ray Chandler, Mar 05 2005

A104393 Sums of 3 distinct positive pentatope numbers (A000332).

Original entry on oeis.org

21, 41, 51, 55, 76, 86, 90, 106, 110, 120, 132, 142, 146, 162, 166, 176, 197, 201, 211, 216, 226, 230, 231, 246, 250, 260, 281, 285, 295, 315, 336, 337, 341, 346, 350, 351, 366, 370, 371, 380, 401, 405, 406, 415, 435, 457, 461, 471, 491, 501

Offset: 0

Jonathan Vos Post, Mar 05 2005

Pentatope number Ptop(n) = binomial(n+3,4) = n*(n+1)*(n+2)*(n+3)/24. Hyun Kwang Kim asserts that every positive integer can be represented as the sum of no more than 8 pentatope numbers; but in this sequence we are only concerned with sums of nonzero distinct pentatope numbers.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Pentatope Number.

Cf. A000332, A100009, A102857, A104392.

Total/@Subsets[Table[Binomial[n+3,4],{n,10}],{3}]//Sort (* Harvey P. Dale, Feb 14 2018 *)

a(n) = Ptop(i) + Ptop(j) + Ptop(k) for some positive i=/=j=/=k and Ptop(n) = binomial(n+3,4).

Extended by Ray Chandler, Mar 05 2005

A145454 Exponential transform of binomial(n,4) = A000332.

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 15, 35, 105, 756, 6510, 46530, 283470, 1667380, 11457446, 99776040, 969295145, 9298091180, 86154691680, 804769174536, 8052676029420, 88489327173660, 1038440150703340, 12501684521410700, 151866259113256611

Offset: 0

Alois P. Heinz, Oct 10 2008

nonn

a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 4 balls are seen at the top.

a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 4 labels.

Alois P. Heinz, Table of n, a(n) for n = 0..500
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

4th column of A145460, A143398.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1) *binomial(j,4) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);

Table[Sum[BellY[n, k, Binomial[Range[n], 4]], {k, 0, n}], {n, 0, 25}] (* Vladimir Reshetnikov, Nov 09 2016 *)

E.g.f.: exp(exp(x)*x^4/4!).

A104394 Sums of 4 distinct positive pentatope numbers (A000332).

Original entry on oeis.org

56, 91, 111, 121, 125, 147, 167, 177, 181, 202, 212, 216, 231, 232, 236, 246, 251, 261, 265, 286, 296, 300, 316, 320, 330, 342, 351, 352, 356, 371, 372, 376, 381, 385, 386, 406, 407, 411, 416, 420, 421, 436, 440, 441, 450, 462, 472, 476, 492, 496

Offset: 1

Jonathan Vos Post, Mar 05 2005

Pentatope number Ptop(n) = binomial(n+3,4) = n*(n+1)*(n+2)*(n+3)/24. Hyun Kwang Kim asserts that every positive integer can be represented as the sum of no more than 8 pentatope numbers; but in this sequence we are only concerned with sums of nonzero distinct pentatope numbers.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Pentatope Number.

Cf. A000332, A100009, A102857, A104392, A104393.

a(n) = Ptop(h) + Ptop(i) + Ptop(j) + Ptop(k) for some positive h=/=i=/=j=/=k and Ptop(n) = binomial(n+3,4).

Extended by Ray Chandler, Mar 05 2005

A104395 Sums of 5 distinct positive pentatope numbers (A000332).

Original entry on oeis.org

126, 182, 217, 237, 247, 251, 266, 301, 321, 331, 335, 357, 377, 386, 387, 391, 412, 421, 422, 426, 441, 442, 446, 451, 455, 456, 477, 497, 507, 511, 532, 542, 546, 551, 561, 562, 566, 576, 581, 586, 591, 595, 606, 616, 620, 626, 630, 642, 646, 650

Offset: 1

Jonathan Vos Post, Mar 05 2005

Hyun Kwang Kim asserts that every positive integer can be represented as the sum of no more than 8 pentatope numbers; but in this sequence we are only concerned with sums of nonzero distinct pentatope numbers.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.

Robert Israel, Table of n, a(n) for n = 1..10000
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
Eric Weisstein's World of Mathematics, Pentatope Number.

Cf. A000332, A100009, A102857, A104392, A104393, A104394.

N:= 1000: # for terms <= N
ptop:= n -> n*(n+1)*(n+2)*(n+3)/24:
P:= 1:
for i from 1 while ptop(i) < N do
  P:= P * (1 + x*y^ptop(i))
od:
sort(map(degree,convert(convert(series(coeff(P,x,5),y,N+1),polynom),list)));
# Robert Israel, Nov 20 2023

a(n) = Ptop(g) + Ptop(h) + Ptop(i) + Ptop(j) + Ptop(k) for some positive g=/=h=/=i=/=j=/=k and Ptop(n) = binomial(n+3,4).

Extended by Ray Chandler, Mar 05 2005

A342865 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1234. 0 <= k <= A000332(n).

Original entry on oeis.org

1, 1, 2, 6, 23, 1, 103, 12, 4, 0, 0, 1, 513, 102, 63, 10, 6, 12, 8, 0, 0, 5, 0, 0, 0, 0, 0, 1, 2761, 770, 665, 196, 146, 116, 142, 46, 10, 72, 32, 24, 0, 13, 0, 12, 18, 0, 0, 10, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Peter Kagey, Mar 26 2021

Equivalently the table for the pattern 4321.

First column is A005802.

			Table begins:
  n\k|       0        1        2        3        4        5        6
  ---+-------------------------------------------------------------------
   0 |       1;
   1 |       1;
   2 |       2;
   3 |       6;
   4 |      23,       1;
   5 |     103,      12,       4,       0,       0,       1;
   6 |     513,     102,      63,      10,       6,      12,       8, ...
   7 |    2761,     770,     665,     196,     146,     116,     142, ...
   8 |   15767,    5545,    5982,    2477,    2148,    1204,    1782, ...
   9 |   94359,   39220,   49748,   25886,   25190,   13188,   19936, ...
  10 |  586590,  276144,  396642,  244233,  260505,  142550,  210663, ...
  11 | 3763290, 1948212, 3089010, 2167834, 2493489, 1476655, 2136586, ...

Peter Kagey, Rows n = 0..13, flattened, based on Anders Kaseorg's Rust program in the Code Golf Stack Exchange link.
Anders Kaseorg, Answer: Patterns in Permutations, Code Golf Stack Exchange.

Cf. A000332, A005802.

Analogous for other patterns: A008302 (12), A138159 (321), A263771 (312), A342840 (1342), A342860 (2413), A342861 (1324), A342862 (2143), A342863 (1243), A342864 (1432).

A145919 A000332(n) = a(n)(3a(n) - 1)/2.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, -3, 5, 7, -9, 12, 15, -18, 22, 26, -30, 35, 40, -45, 51, 57, -63, 70, 77, -84, 92, 100, -108, 117, 126, -135, 145, 155, -165, 176, 187, -198, 210, 222, -234, 247, 260, -273, 287, 301, -315, 330, 345, -360, 376, 392, -408, 425, 442, -459, 477

Offset: 0

Matthew Vandermast, Oct 28 2008

As the formula in the description shows, all members of A000332 belong to the generalized pentagonal sequence (A001318). A001318 also lists all nonnegative numbers that belong to A145919.

			a(6) = -3 and A000332(6) = (-3)(-10)/2 = 15.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pentagonal Number.
Eric Weisstein's World of Mathematics, Pentatope Number.
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Cf. A000326, A145920.

CoefficientList[Series[(-x^4*(x^4 + 2*x^3 - 3*x^2 + 2*x + 1))/((x - 1)^3*(1 + x^2 + x)^3), {x,0,50}], x] (* G. C. Greubel, Jun 13 2017 *)
LinearRecurrence[{0,0,3,0,0,-3,0,0,1},{0,0,0,0,1,2,-3,5,7},60] (* Harvey P. Dale, Feb 13 2023 *)

my(x='x+O('x^50)); concat([0, 0, 0, 0], Vec((-x^4*(x^4 +2*x^3 -3*x^2 +2*x +1))/((x-1)^3*(1+x^2+x)^3))) \\ G. C. Greubel, Jun 13 2017

a(n+3) = A001840(n) when 3 does not divide n, A001840(n)*-1 otherwise.
After first two zeros, this sequence consists of all values of A001318(n) and A045943(n)*(-1), n>=0, sorted in order of increasing absolute value.
G.f.: (-x^4*(x^4+2*x^3-3*x^2+2*x+1))/((x-1)^3*(1+x^2+x)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
a(n) = ((n^2-3*n+1)*(1-4*cos(2*Pi*n/3))+3)/18. - Natalia L. Skirrow, Apr 14 2025

A145920 List of numbers that are both pentagonal (A000326) and binomial coefficients C(n,4) (A000332).

Original entry on oeis.org

0, 1, 5, 35, 70, 210, 330, 715, 1001, 1820, 2380, 3876, 4845, 7315, 8855, 12650, 14950, 20475, 23751, 31465, 35960, 46376, 52360, 66045, 73815, 91390, 101270, 123410, 135751, 163185, 178365, 211876, 230300, 270725, 292825, 341055, 367290, 424270

Offset: 1

Matthew Vandermast, Oct 28 2008

All binomial coefficients C(n,4) belong to the generalized pentagonal sequence (A001318).

Pentagonal numbers of generalized pentagonal number (A001318) index number. - Raphie Frank, Nov 25 2012

			35, for example, is both A000326(5) and A000332(7).

William A. Tedeschi, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentagonal Number.
Eric Weisstein's World of Mathematics, Pentatope Number.

Cf. A141919, of which this is a subsequence.

a(n+1) = A000326 (A001318(n)).
Positive values of A000332(n) belong to the sequence if and only if 3 does not divide n. A000332(n) is positive when n>3.
Conjecture: a(n) = a(n-1) + 4a(n-2) - 4a(n-3) - 6a(n-4) + 6a(n-5) + 4a(n-6) - 4a(n-7) - a(n-8) + a(n-9). - R. J. Mathar, Oct 29 2008
Conjecture: G.f.: x^2(1 + 4x + 26x^2 + 19x^3 + 4x^5 + x^6 + 26x^4)/((1+x)^4(1-x)^5). - R. J. Mathar, Oct 29 2008
a(n) = (27x^4 - 18x^3 - 3x^2 + 2x)/8 where x = floor(n/2)*(-1)^n, for n >= 1. - William A. Tedeschi, Aug 16 2010

A104396 Sums of 6 distinct positive pentatope numbers (A000332).

Original entry on oeis.org

252, 336, 392, 427, 447, 456, 457, 461, 512, 547, 567, 577, 581, 596, 621, 631, 651, 661, 665, 677, 687, 707, 712, 717, 721, 732, 742, 746, 752, 756, 761, 772, 776, 786, 796, 816, 826, 830, 841, 852, 872, 881, 882, 886, 897, 907, 916, 917, 921, 932

Offset: 1

Jonathan Vos Post, Mar 05 2005

Pentatope number Ptop(n) = binomial(n+3,4) = n*(n+1)*(n+2)*(n+3)/24.

Hyun Kwang Kim asserts that every positive integer can be represented as the sum of no more than 8 pentatope numbers; but in this sequence we are only concerned with sums of nonzero distinct pentatope numbers.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Pentatope Number.

Cf. A000332, A100009, A102857, A104392, A104393, A104394, A104395.

a(n) = Ptop(f) + Ptop(g) + Ptop(h) + Ptop(i) + Ptop(j) + Ptop(k) for some positive f=/=g=/=h=/=i=/=j=/=k and Ptop(n) = binomial(n+3,4).

Extended by Ray Chandler, Mar 05 2005

cp's OEIS Frontend

A000391 Euler transform of A000332.

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A104392 Sums of 2 distinct positive pentatope numbers (A000332).

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A104393 Sums of 3 distinct positive pentatope numbers (A000332).

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A145454 Exponential transform of binomial(n,4) = A000332.

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A104394 Sums of 4 distinct positive pentatope numbers (A000332).

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A104395 Sums of 5 distinct positive pentatope numbers (A000332).

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A342865 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1234. 0 <= k <= A000332(n).

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A145919 A000332(n) = a(n)*(3*a(n) - 1)/2.

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A145919 A000332(n) = a(n)(3a(n) - 1)/2.