A060354 -id:A060354 - cp's OEIS frontend

A337762 Number of partitions of the n-th n-gonal number into n-gonal numbers.

1, 1, 2, 4, 8, 21, 56, 144, 370, 926, 2275, 5482, 12966, 30124, 68838, 154934, 343756, 752689, 1627701, 3479226, 7355608, 15390682, 31889732, 65465473, 133212912, 268811363, 538119723, 1069051243, 2108416588, 4129355331, 8033439333

Offset: 0

Ilya Gutkovskiy, Sep 19 2020

nonn

			a(3) = 4 because the third triangular number is 6 and we have [6], [3, 3], [3, 1, 1, 1] and [1, 1, 1, 1, 1, 1].

David A. Corneth, Table of n, a(n) for n = 0..278 (first 51 terms from Vaclav Kotesovec)
Eric Weisstein's World of Mathematics, Polygonal Number
Index entries for sequences related to partitions
Index to sequences related to polygonal numbers

Cf. A000041, A037444, A060354, A072964, A337763, A337764.

nmax = 20; Table[SeriesCoefficient[Product[1/(1 - x^(k*((k*(n - 2) - n + 4)/2))), {k, 1, n}], {x, 0, n*(4 - 3*n + n^2)/2}], {n, 0, nmax}] (* Vaclav Kotesovec, Sep 19 2020 *)

a(n) = [x^p(n,n)] Product_{k=1..n} 1 / (1 - x^p(n,k)), where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.

A004255 n(n+1)(n^2 -3n + 6)/8.

Original entry on oeis.org

1, 3, 9, 25, 60, 126, 238, 414, 675, 1045, 1551, 2223, 3094, 4200, 5580, 7276, 9333, 11799, 14725, 18165, 22176, 26818, 32154, 38250, 45175, 53001, 61803, 71659, 82650, 94860, 108376, 123288, 139689, 157675, 177345, 198801, 222148, 247494, 274950

Offset: 1

Dennis S. Kluk (mathemagician(AT)ameritech.net)

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Partial sums of A060354. Equals (1/2) A062026.

[n*(n+1)*(n^2-3*n+6)/8: n in [1..50]]; // Vincenzo Librandi, Jun 07 2013

lst={};Do[AppendTo[lst, n*(n+1)*(n^2-3*n+6)/8], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 19 2008 *)
Table[n(n+1)(n^2-3n+6)/8,{n,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,3,9,25,60},40] (* Harvey P. Dale, May 27 2023 *)

a(n)=n*(n+1)*(n^2-3*n+6)/8 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: -x*(1-2*x+4*x^2) / (x-1)^5. - Simon Plouffe in his 1992 dissertation.

A292551 Expansion of x(1 - 2x + x^2 + 7x^3 - x^4)/((1 - x)^4(1 + x)^3).

Original entry on oeis.org

0, 1, -1, 3, 4, 12, 21, 34, 56, 75, 115, 141, 204, 238, 329, 372, 496, 549, 711, 775, 980, 1056, 1309, 1398, 1704, 1807, 2171, 2289, 2716, 2850, 3345, 3496, 4064, 4233, 4879, 5067, 5796, 6004, 6821, 7050, 7960, 8211, 9219, 9493, 10604, 10902, 12121, 12444, 13776, 14125, 15575

Offset: 0

Ilya Gutkovskiy, Sep 18 2017

The n-th generalized n-gonal number (for n >= 5).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A006484, A060354, A100119, A195152.

Main diagonal of A303301.

List([0..50],n->(2*n^3-2*n^2+(-1)^n*(2*n^2-11*n-6)-5*n+6)/16); # Muniru A Asiru, Aug 08 2018

a:= n-> (m-> m*((n-2)*m-(n-4))/2)(-ceil(n/2)*(-1)^n):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 29 2018

CoefficientList[Series[x (1 - 2 x + x^2 + 7 x^3 - x^4)/((1 - x)^4 (1 + x)^3), {x, 0, 50}], x]
Table[SeriesCoefficient[x (1 + (n - 4) x + x^2)/((1 - x)^3 (1 + x)^2), {x, 0, n}], {n, 0, 50}]
LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {0, 1, -1, 3, 4, 12, 21}, 51]
Table[(2 n^3 - 2 n^2 + (-1)^n (2 n^2 - 11 n - 6) - 5 n + 6)/16, {n, 0, 50}]

x='x+O('x^99); concat(0, Vec(x*(1-2*x+x^2+7*x^3-x^4)/((1-x)^4*(1+x)^3))) \\ Altug Alkan, Sep 18 2017

G.f.: x*(1 - 2*x + x^2 + 7*x^3 - x^4)/((1 - x)^4*(1 + x)^3).
a(n) = [x^n] x*(1 + (n - 4)*x + x^2)/((1 - x)^3*(1 + x)^2).
E.g.f.: (1/16)*((-6 + 9*x + 2*x^2)*exp(-x) + (6 - 5*x + 4*x^2 + 2*x^3)*exp(x)).
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7).
a(n) = (2*n^3 - 2*n^2 + (-1)^n*(2*n^2 - 11*n - 6) - 5*n + 6)/16.

A317302 Square array T(n,k) = (n - 2)(k - 1)k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, -3, 0, 1, 2, 0, -8, 0, 1, 3, 3, -2, -15, 0, 1, 4, 6, 4, -5, -24, 0, 1, 5, 9, 10, 5, -9, -35, 0, 1, 6, 12, 16, 15, 6, -14, -48, 0, 1, 7, 15, 22, 25, 21, 7, -20, -63, 0, 1, 8, 18, 28, 35, 36, 28, 8, -27, -80, 0, 1, 9, 21, 34, 45, 51, 49, 36, 9, -35, -99, 0, 1, 10, 24, 40, 55, 66

Offset: 0

Omar E. Pol, Aug 09 2018

Note that the formula gives several kinds of numbers, for example:
Row 0 gives 0 together with A258837.
Row 1 gives 0 together with A080956.
Row 2 gives A001477, the nonnegative numbers.
For n >= 3, row n gives the n-gonal numbers (see Crossrefs section).

			Array begins:
------------------------------------------------------------------------
n\k  Numbers       Seq. No.   0   1   2   3   4    5    6    7    8
------------------------------------------------------------------------
0    ............ (A258837):  0,  1,  0, -3, -8, -15, -24, -35, -48, ...
1    ............ (A080956):  0,  1,  1,  0, -2,  -5,  -9, -14, -20, ...
2    Nonnegatives  A001477:   0,  1,  2,  3,  4,   5,   6,   7,   8, ...
3    Triangulars   A000217:   0,  1,  3,  6, 10,  15,  21,  28,  36, ...
4    Squares       A000290:   0,  1,  4,  9, 16,  25,  36,  49,  64, ...
5    Pentagonals   A000326:   0,  1,  5, 12, 22,  35,  51,  70,  92, ...
6    Hexagonals    A000384:   0,  1,  6, 15, 28,  45,  66,  91, 120, ...
7    Heptagonals   A000566:   0,  1,  7, 18, 34,  55,  81, 112, 148, ...
8    Octagonals    A000567:   0,  1,  8, 21, 40,  65,  96, 133, 176, ...
9    9-gonals      A001106:   0,  1,  9, 24, 46,  75, 111, 154, 204, ...
10   10-gonals     A001107:   0,  1, 10, 27, 52,  85, 126, 175, 232, ...
11   11-gonals     A051682:   0,  1, 11, 30, 58,  95, 141, 196, 260, ...
12   12-gonals     A051624:   0,  1, 12, 33, 64, 105, 156, 217, 288, ...
13   13-gonals     A051865:   0,  1, 13, 36, 70, 115, 171, 238, 316, ...
14   14-gonals     A051866:   0,  1, 14, 39, 76, 125, 186, 259, 344, ...
15   15-gonals     A051867:   0,  1, 15, 42, 82, 135, 201, 280, 372, ...
...

Omar E. Pol, Polygonal numbers.
The OEIS, Polygonal numbers.
University of Surrey, Dept. of Mathematics, Polygonal Numbers - or Numbers as Shapes.
Eric Weisstein's World of Mathematics, Figurate Number.
Eric Weisstein's World of Mathematics, Polygonal Number.
Wikipedia, Polygonal number.
Index to sequences related to polygonal numbers

Column 0 gives A000004.
Column 1 gives A000012.
Column 2 gives A001477, which coincides with the row numbers.
Main diagonal gives A060354.
Row 0 gives 0 together with A258837.
Row 1 gives 0 together with A080956.
Row 2 gives A001477, the same as column 2.
For n >= 3, row n gives the n-gonal numbers: A000217 (n=3), A000290 (n=4), A000326 (n=5), A000384 (n=6), A000566 (n=7), A000567 (n=8), A001106 (n=9), A001107 (n=10), A051682 (n=11), A051624 (n=12), A051865 (n=13), A051866 (n=14), A051867 (n=15), A051868 (n=16), A051869 (n=17), A051870 (n=18), A051871 (n=19), A051872 (n=20), A051873 (n=21), A051874 (n=22), A051875 (n=23), A051876 (n=24), A255184 (n=25), A255185 (n=26), A255186 (n=27), A161935 (n=28), A255187 (n=29), A254474 (n=30).
Cf. A139600, A139601.
Cf. A303301 (similar table but with generalized polygonal numbers).

T(n,k) = A139600(n-2,k) if n >= 2.

T(n,k) = A139601(n-3,k) if n >= 3.

A337763 Number of partitions of the n-th n-gonal number into distinct n-gonal numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 4, 4, 2, 2, 4, 7, 8, 5, 6, 14, 6, 13, 23, 16, 19, 32, 34, 48, 56, 62, 73, 137, 126, 203, 257, 256, 409, 503, 612, 794, 1097, 1203, 1737, 2141, 2773, 3322, 4527, 5087, 7497, 8214, 11238, 12598

Offset: 0

Ilya Gutkovskiy, Sep 19 2020

nonn

			a(5) = 2 because 5th pentagonal number is 35 and we have [35] and [22, 12, 1].

Eric Weisstein's World of Mathematics, Polygonal Number
Index entries for sequences related to partitions
Index to sequences related to polygonal numbers

Cf. A000009, A030273, A060354, A288126, A337762, A337764.

a(n) = [x^p(n,n)] Product_{k=1..n} (1 + x^p(n,k)), where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.

A337764 Number of compositions (ordered partitions) of the n-th n-gonal number into n-gonal numbers.

Original entry on oeis.org

1, 1, 2, 7, 124, 14371, 12842911, 103590035354, 8621925847489749, 8307493939404888703058, 102488432265617100812550713499, 17706351554929677399562928448484650120, 46435685450659378932235460132506329282776942795

Offset: 0

Ilya Gutkovskiy, Sep 19 2020

nonn

			a(3) = 7 because the third triangular number is 6 and we have [6], [3, 3], [3, 1, 1, 1], [1, 3, 1, 1], [1, 1, 3, 1], [1, 1, 1, 3] and [1, 1, 1, 1, 1, 1].

Eric Weisstein's World of Mathematics, Polygonal Number
Index entries for sequences related to compositions
Index to sequences related to polygonal numbers

Cf. A011782, A060354, A224366, A224677, A337762, A337763.

a(n) = [x^p(n,n)] 1 / (1 - Sum_{k=1..n} x^p(n,k)), where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.

A360855 Array read by antidiagonals: T(m,n) is the number of triangles in the rook graph K_m X K_n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 4, 2, 2, 4, 10, 8, 6, 8, 10, 20, 20, 16, 16, 20, 20, 35, 40, 35, 32, 35, 40, 35, 56, 70, 66, 60, 60, 66, 70, 56, 84, 112, 112, 104, 100, 104, 112, 112, 84, 120, 168, 176, 168, 160, 160, 168, 176, 168, 120, 165, 240, 261, 256, 245, 240, 245, 256, 261, 240, 165

Offset: 1

Andrew Howroyd, Feb 24 2023

A triangle is a clique of size 3. Also, a 3-cycle.

			Array begins:
=======================================
m\n|  1   2   3   4   5   6   7   8 ...
---+-----------------------------------
1  |  0   0   1   4  10  20  35  56 ...
2  |  0   0   2   8  20  40  70 112 ...
3  |  1   2   6  16  35  66 112 176 ...
4  |  4   8  16  32  60 104 168 256 ...
5  | 10  20  35  60 100 160 245 360 ...
6  | 20  40  66 104 160 240 350 496 ...
7  | 35  70 112 168 245 350 490 672 ...
8  | 56 112 176 256 360 496 672 896 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Eric Weisstein's World of Mathematics, Rook Graph.

Main diagonal is A288961.
Rows n=1..3 are A000292(n-2), A007290, A060354.
Cf. A286418, A360853.

T(m, n) = m*binomial(n,3) + n*binomial(m,3)

T(m,n) = m*binomial(n,3) + n*binomial(m,3).

T(m,n) = T(n,m).

A383778 a(n) = n(n^2 - 3n + 10)*2^(n-4).

Original entry on oeis.org

0, 1, 4, 15, 56, 200, 672, 2128, 6400, 18432, 51200, 137984, 362496, 931840, 2351104, 5836800, 14286848, 34537472, 82575360, 195493888, 458752000, 1067974656, 2468347904, 5667553280, 12935233536, 29360128000, 66303557632, 149032009728, 333531054080, 743431995392

Offset: 0

Enrique Navarrete, May 09 2025

a(n) is the number of strings of length n defined on {0,1,2,3} that contain exactly one 3, zero or two 2s and have no restriction on the number of 0s and 1s.

			a(3) = 15 since the strings are 322 (3 of this type), 300 (3 of this type), 311 (3 of this type), and 301 (6 of this type).

Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A060354.

a[n_] := n*(n^2-3*n+10)*2^(n-4); Array[a, 30, 0] (* Amiram Eldar, May 09 2025 *)

E.g.f.: x*(1 + x^2/2)*exp(2*x).

G.f.: x*(1 - 4*x + 7*x^2)/(1 - 2*x)^4. - Stefano Spezia, May 10 2025

A218152 a(n) = 1 + n + ((n-1)*n^2)/2.

Original entry on oeis.org

1, 2, 5, 13, 29, 56, 97, 155, 233, 334, 461, 617, 805, 1028, 1289, 1591, 1937, 2330, 2773, 3269, 3821, 4432, 5105, 5843, 6649, 7526, 8477, 9505, 10613, 11804, 13081, 14447, 15905, 17458, 19109, 20861, 22717, 24680, 26753, 28939, 31241, 33662, 36205, 38873

Offset: 0

Mokhtar Mohamed, Oct 24 2012

a(n) = sum(i=0,1,2,...k) d(i)*C(n,i), d(0)=a(0), C(n,i)=0 for all i > n. I would introduce the arithmetic-arithmetic sequence which is defined as the sequence of finite differences, that is, with k consecutive rows of differences, whose first terms are d(1), d(2), d(3),..., d(k), the last row (k-th row) being of a constant difference. Here, it is submitted a special case of the above mentioned sequence with k=3, d(0)=d(1)=1,  d(2)=2, d(3)=3.
This sequence is not in Comtet. - T. D. Noe, Nov 16 2012
a(n) appears to be the number of configurations of n equilateral triangles that are allowed to have common vertices, where A002061(n) gives the number of connected configurations and A060354(n) is the number of configurations consisting of several pieces. - Anton Zakharov, May 13 2018

			for n=5, a(5) = 1+5+(4*25)/2 = 1+5+100/2 = 1+5+50 = 56.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72.

Colin Barker, Table of n, a(n) for n = 0..1000
Anton Zakharov, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000027, A000124, A000125, A060354, A002061.

Table[1+n+((n-1)n^2)/2,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,2,5,13},50] (* Harvey P. Dale, May 04 2023 *)

Vec((1 - 2*x + 3*x^2 + x^3) / (1 - x)^4 + O(x^40)) \\ Colin Barker, May 13 2018

a(n) = a(n-1)+(4-5*n+3*n^2)/2 for n > 0 and a(0)=1.
a(n) = A006000(n-1)+1 for n > 0. - Antti Karttunen, Oct 24 2012
a(n) = A060354(n) + A002061(n). - Anton Zakharov, May 13 2018
G.f.: (x^3+3*x^2-2*x+1)/(x-1)^4. - Alois P. Heinz, May 13 2018
From Colin Barker, May 13 2018: (Start)
a(n) = (2 + 2*n - n^2 + n^3) / 2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.
(End)

Corrected and edited by Mokhtar Mohamed, Nov 17 2012

Missing term 1937 inserted by Alois P. Heinz, Jun 11 2017

A294958 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*((k-2)^2+k)/2).

Original entry on oeis.org

1, 1, 3, 9, 28, 75, 198, 494, 1243, 3061, 7500, 18055, 43057, 101292, 236178, 545218, 1248480, 2835059, 6390360, 14298631, 31778782, 70168935, 153993321, 335977369, 728962258, 1573189113, 3377881482, 7217395643, 15348900996, 32494548816, 68494383520, 143773075158, 300568066729

Offset: 0

Ilya Gutkovskiy, Nov 12 2017

nonn

Euler transform of A060354.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index to sequences related to polygonal numbers

Cf. A057145, A060354, A294959.

N:=100:
S:= series(mul(1/(1 - x^k)^(k*((k-2)^2+k)/2),k=1..N),x,N+1):
seq(coeff(S,x,k),k=0..N); # Robert Israel, Nov 12 2017

nmax = 32; CoefficientList[Series[Product[1/(1 - x^k)^(k ((k - 2)^2 + k)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 ((d - 2)^2 + d)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A060354(k).

a(n) ~ exp(2*Zeta'(-1) + 3*Zeta(3) / (8*Pi^2) - Pi^16 / (1036800000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (36000 * Zeta(5)^2) - 2*Zeta(3)^2 / (15*Zeta(5)) + Zeta'(-3)/2 + (-Pi^12 / (3600000 * 2^(2/5) * 3^(1/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (150 * 2^(2/5) * 3^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (12000 * 2^(4/5) * 3^(2/5) * Zeta(5)^(7/5)) + 2^(1/5) * Zeta(3) / (3*Zeta(5))^(2/5)) * n^(2/5) - (Pi^4 / (60 * 2^(1/5) * (3*Zeta(5))^(3/5))) * n^(3/5) + (5*(3*Zeta(5))^(1/5) / 2^(8/5)) * n^(4/5)) * (3*Zeta(5))^(53/400) / (2^(47/200) * sqrt(5*Pi) * n^(253/400)). - Vaclav Kotesovec, Nov 12 2017

cp's OEIS Frontend

A337762 Number of partitions of the n-th n-gonal number into n-gonal numbers.

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A004255 n(n+1)(n^2 -3n + 6)/8.

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Magma

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A292551 Expansion of x*(1 - 2*x + x^2 + 7*x^3 - x^4)/((1 - x)^4*(1 + x)^3).

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A317302 Square array T(n,k) = (n - 2)*(k - 1)*k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards.

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A337763 Number of partitions of the n-th n-gonal number into distinct n-gonal numbers.

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A337764 Number of compositions (ordered partitions) of the n-th n-gonal number into n-gonal numbers.

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Author

Keywords

Examples

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Crossrefs

Formula

A360855 Array read by antidiagonals: T(m,n) is the number of triangles in the rook graph K_m X K_n.

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A383778 a(n) = n*(n^2 - 3*n + 10)*2^(n-4).

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A292551 Expansion of x(1 - 2x + x^2 + 7x^3 - x^4)/((1 - x)^4(1 + x)^3).

A317302 Square array T(n,k) = (n - 2)(k - 1)k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards.

A383778 a(n) = n(n^2 - 3n + 10)*2^(n-4).