A072964 -id:A072964 - cp's OEIS frontend

A073420 G.f.: 1/B(x) where B(x) = g.f. for A072964.

1, -1, -1, -1, 0, -2, -4, -4, -8, -10, -17, -10, 6, 45, 260, 604, 1285, 3329, 6755, 12550, 25790, 46302, 75922, 129739, 188164, 215880, 177637, -208052, -1480885, -4563499, -11984644, -28275270, -61059278, -126088969, -248020723, -462216478, -823834918, -1383451194, -2136524063

Offset: 0

N. J. A. Sloane, Aug 26 2002

Cf. A072964.

More terms from R. J. Mathar, Feb 13 2008

A321507 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)/2))^A072964(k).

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 10, 10, 10, 22, 29, 29, 56, 70, 70, 127, 176, 176, 283, 367, 395, 644, 833, 889, 1315, 1714, 1910, 2791, 3606, 3942, 5538, 7413, 8169, 11100, 14544, 16140, 21927, 28886, 32344, 42152, 54728, 62624, 81625, 105148, 120310, 152699, 197624

Offset: 0

Ilya Gutkovskiy, Nov 11 2018

nonn

a(n) is the number of partitions of n into triangular numbers k*(k + 1)/2 of A072964(k) kinds.

			a(6) = 10 because we have [{6}], [{3, 3}],  [{3}, {3}], [{3, 1, 1, 1}], [{3}, {1, 1, 1}], [{3}, {1}, {1}, {1}], [{1, 1, 1, 1, 1, 1}], [{1, 1, 1}, {1, 1, 1}], [{1, 1, 1}, {1}, {1}, {1}] and [{1}, {1}, {1}, {1}, {1}, {1}].

Index entries for sequences related to partitions

Cf. A000217, A001970, A007294, A072964, A300300.

b[n_] := b[n] = SeriesCoefficient[Product[1/(1 - x^(k (k + 1)/2)), {k, 1, n}], {x, 0, n (n + 1)/2}]; a[n_] := a[n] = SeriesCoefficient[Product[1/(1 - x^(k (k + 1)/2))^b[k], {k, 1, n}], {x, 0, n}]; Table[a[n], {n, 0, 46}]

G.f.: Product_{k>=1} 1/(1 - x^A000217(k))^A007294(A000217(k)).

A037444 Number of partitions of n^2 into squares.

Original entry on oeis.org

1, 1, 2, 4, 8, 19, 43, 98, 220, 504, 1116, 2468, 5368, 11592, 24694, 52170, 108963, 225644, 462865, 941528, 1899244, 3801227, 7550473, 14889455, 29159061, 56722410, 109637563, 210605770, 402165159, 763549779, 1441686280, 2707535748, 5058654069, 9404116777

Offset: 0

Wouter Meeussen

Is lim_{n->inf} a(n)^(1/n) > 1? - Paul D. Hanna, Aug 20 2002

The limit above is equal to 1 (see formula by Hardy & Ramanujan for A001156). - Vaclav Kotesovec, Dec 29 2016

T. D. Noe, Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..945 (terms n = 0..100 from T. D. Noe, terms n = 101..500 from Alois P. Heinz)
J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301.
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.

Entries with square index in A001156.
Cf. A072964, A030273, A000041, A000290, A229239, A229468.
Cf. A003108, A046042.
Cf. A259792, A259793.
A row or column of the array in A259799.

a037444 n = p (map (^ 2) [1..]) (n^2) where
   p _      0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 14 2011

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i)))
    end:
a:= n-> b(n^2, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 15 2013

max=33; se = Series[ Product[1/(1-x^(k^2)), {k, 1, max}], {x, 0, max^2}]; a[n_] := Coefficient[se, x^(n^2)]; a[0] = 1; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Oct 18 2011 *)

a(n) = A001156(n^2) = coefficient of x^(n^2) in the series expansion of Prod_{k>=1} 1/(1 - x^(k^2)).

a(n) ~ 3^(-1/2) * (4*Pi)^(-7/6) * Zeta(3/2)^(2/3) * n^(-7/3) * exp(2^(-4/3) * 3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(2/3)) [Hardy & Ramanujan, 1917, modified from A001156]. - Vaclav Kotesovec, Dec 29 2016

A288126 Number of partitions of n-th triangular number (A000217) into distinct triangular parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 2, 4, 7, 6, 4, 14, 15, 19, 31, 28, 43, 57, 80, 103, 127, 181, 234, 295, 398, 539, 663, 888, 1178, 1419, 1959, 2519, 3102, 4201, 5282, 6510, 8717, 11162, 13557, 18108, 22965, 28206, 36860, 46350, 58060, 73857, 93541, 117058, 147376, 186158, 232949, 292798, 365639

Offset: 0

Ilya Gutkovskiy, Jun 05 2017

nonn

			a(4) = 2 because 4th triangular number is 10 and we have [10], [6, 3, 1].

Robert Israel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Triangular Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A000217, A007294, A024940, A030273, A037444, A072964.

N:= 100:
G:= mul(1+x^(k*(k+1)/2),k=1..N):
seq(coeff(G,x,n*(n+1)/2),n=0..N); # Robert Israel, Jun 06 2017

Table[SeriesCoefficient[Product[1 + x^(k (k + 1)/2), {k, 1, n}], {x, 0, n (n + 1)/2}], {n, 0, 54}]

a(n) = [x^(n*(n+1)/2)] Product_{k>=1} (1 + x^(k(k+1)/2)).

a(n) = A024940(A000217(n)).

A298858 Number of ordered ways of writing n-th triangular number as a sum of n nonzero triangular numbers.

Original entry on oeis.org

1, 1, 0, 0, 4, 11, 86, 777, 4670, 36075, 279482, 2345201, 21247326, 197065752, 1983741228, 20769081251, 228078253168, 2604226354265, 30880251148086, 379415992755572, 4818158748326064, 63116999199457944, 851467484377802094, 11811530978240316682, 168243449082524484856

Offset: 0

Ilya Gutkovskiy, Jan 27 2018

nonn

			a(4) = 4 because fourth triangular number is 10 and we have [3, 3, 3, 1], [3, 3, 1, 3], [3, 1, 3, 3] and [1, 3, 3, 3].

Eric Weisstein's World of Mathematics, Triangular Number
Index to sequences related to polygonal numbers

Cf. A000217, A007294, A023361, A024940, A072964, A106337, A196010, A288126, A298329, A298330.

Table[SeriesCoefficient[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^n, {x, 0, n (n + 1)/2}], {n, 0, 24}]

a(n) = [x^(n*(n+1)/2)] (Sum_{k>=1} x^(k*(k+1)/2))^n.

A298269 Number of partitions of the n-th tetrahedral number into tetrahedral numbers.

Original entry on oeis.org

1, 1, 2, 4, 11, 29, 94, 304, 1005, 3336, 11398, 38739, 132340, 451086, 1541074, 5242767, 17779666, 60048847, 202124143, 677000711, 2256910444, 7486274436, 24713275977, 81162110629, 265192045408, 862061443031, 2788194736946, 8972104829849, 28726271274133, 91515498561954, 290116750935925

Offset: 0

Ilya Gutkovskiy, Jan 27 2018

nonn

			a(3) = 4 because third tetrahedral number is 10 and we have [10], [4, 4, 1, 1], [4, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

David A. Corneth, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers
Index entries for related partition-counting sequences

Cf. A000292, A037444, A068980, A072964, A298857.

Table[SeriesCoefficient[Product[1/(1 - x^(k (k + 1) (k + 2)/6)), {k, 1, n}], {x, 0, n (n + 1) (n + 2)/6}], {n, 0, 30}]

a(n) = [x^A000292(n)] Product_{k>=1} 1/(1 - x^A000292(k)).

a(n) = A068980(A000292(n)).

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

Original entry on oeis.org

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.

A307614 Number of partitions of the n-th triangular number into consecutive positive triangular numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1

Offset: 1

Ilya Gutkovskiy, Apr 18 2019

nonn

			A000217(4) = 10 = 1 + 3 + 6, so a(4) = 2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000217, A034706, A072964, A196010, A288126, A298858, A307608.

a(n) = [x^(n*(n+1)/2)] Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k*(k+1)/2).

A337797 Number of partitions of the n-th n-gonal pyramidal number into n-gonal pyramidal numbers.

Original entry on oeis.org

1, 1, 2, 4, 13, 45, 198, 858, 3728, 16115, 69125, 292940, 1224628, 5052396, 20570806, 82655098, 327881398, 1284663878, 4973614490, 19034194696, 72037124003, 269723590850, 999517370314, 3667158097572, 13325691939021, 47975192145998

Offset: 0

Ilya Gutkovskiy, Sep 22 2020

nonn

			a(3) = 4 because the third tetrahedral (or triangular pyramidal) number is 10 and we have [10], [4, 4, 1, 1], [4, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

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

Cf. A006484, A072964, A298269, A337762, A337798, A337799.

a(n) = [x^p(n,n)] Product_{k=1..n} 1 / (1 - x^p(n,k)), where p(n,k) = k * (k + 1) * (k * (n - 2) - n + 5) / 6 is the k-th n-gonal pyramidal number.

A299032 Number of ordered ways of writing n-th triangular number as a sum of n squares of positive integers.

Original entry on oeis.org

1, 1, 0, 3, 6, 0, 12, 106, 420, 2718, 18240, 120879, 694320, 5430438, 40668264, 300401818, 2369504386, 19928714475, 174151735920, 1543284732218, 14224347438876, 135649243229688, 1331658133954940, 13369350846412794, 138122850643702056, 1462610254141337590

Offset: 0

Ilya Gutkovskiy, Feb 01 2018

nonn

			a(4) = 6 because fourth triangular number is 10 and we have [4, 4, 1, 1], [4, 1, 4, 1], [4, 1, 1, 4], [1, 4, 4, 1], [1, 4, 1, 4] and [1, 1, 4, 4].

Cf. A000217, A000290, A066535, A072964, A104383, A126683, A196010, A224677, A224679, A278340, A288126, A298330, A298858, A298939, A299031.

b:= proc(n, t) option remember; local i; if n=0 then
      `if`(t=0, 1, 0) elif t<1 then 0 else 0;
      for i while i^2<=n do %+b(n-i^2, t-1) od; % fi
    end:
a:= n-> b(n*(n+1)/2, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 05 2018

Table[SeriesCoefficient[(-1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n (n + 1)/2}], {n, 0, 25}]

a(n) = [x^(n*(n+1)/2)] (Sum_{k>=1} x^(k^2))^n.

cp's OEIS Frontend

A073420 G.f.: 1/B(x) where B(x) = g.f. for A072964.

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A321507 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)/2))^A072964(k).

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A037444 Number of partitions of n^2 into squares.

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A288126 Number of partitions of n-th triangular number (A000217) into distinct triangular parts.

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Maple

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A298858 Number of ordered ways of writing n-th triangular number as a sum of n nonzero triangular numbers.

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A298269 Number of partitions of the n-th tetrahedral number into tetrahedral numbers.

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

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A307614 Number of partitions of the n-th triangular number into consecutive positive triangular numbers.

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A337797 Number of partitions of the n-th n-gonal pyramidal number into n-gonal pyramidal numbers.