A030273 -id:A030273 - cp's OEIS frontend

A037444 Number of partitions of n^2 into squares.

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

A183953 T(n,k) is the number of strings of numbers x(i=1..n) in 0..k with sum i^2x(i) equal to kn^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 3, 4, 5, 1, 1, 2, 4, 8, 10, 7, 2, 1, 2, 6, 14, 27, 26, 10, 1, 1, 3, 7, 21, 53, 78, 61, 20, 3, 1, 3, 9, 32, 94, 180, 219, 147, 37, 3, 1, 3, 12, 48, 161, 398, 656, 649, 339, 77, 4, 1, 3, 14, 61, 259, 770, 1613, 2195, 1805, 771, 118, 2, 1, 4, 17

Offset: 1

R. H. Hardin, Jan 08 2011

T(n,k) is the number of integer lattice points in k*C(n) where C(n) is the polytope in R^n defined by the equation Sum_{1<=i<=n} i^2*x_i = n^2 and the inequalities 0 <= x_i <= 1. The vertices of the polytope have rational coordinates. Thus row n of the table is an Ehrhart quasi-polynomial of degree n-1. - Robert Israel, Jul 10 2019

			Table starts
.1..1...1....1.....1.....1......1......1.......1.......1.......1........1
.1..1...1....2.....2.....2......2......3.......3.......3.......3........4
.1..2...2....3.....4.....6......7......9......12......14......17.......19
.1..1...4....8....14....21.....32.....48......61......82.....108......139
.2..5..10...27....53....94....161....259.....399.....578.....811.....1120
.1..7..26...78...180...398....770...1387....2330....3738....5772.....8599
.2.10..61..219...656..1613...3539...7099...13225...23247...38938....62599
.1.20.147..649..2195..6301..15601..34847...71509..137520..249799...433038
.3.37.339.1805..7250.23611..65909.163588..369777..775045.1525468..2847243
.3.77.771.4987.23044.85595.268008.737538.1830390.4178324.8894137.17852441
Some solutions for n=5
..4....1....3....0....4....4....0....3....1....3....0....0....0....2....1....0
..3....2....1....0....3....3....0....1....2....1....4....4....0....4....2....4
..3....0....2....1....2....4....4....3....1....4....2....1....0....1....3....4
..2....1....0....1....1....3....4....1....2....2....1....0....0....3....4....3
..1....3....3....3....2....0....0....2....2....1....2....3....4....1....0....0

R. H. Hardin, Table of n, a(n) for n = 1..806

Column 1 is A030273. A183946 (column 2), A183947 (column 3), A183954 (row 3), A183955 (row 4).

A183953rec := proc(n,k,s)
    option remember;
    local c;
    if s < 0 then
        return 0 ;
    elif n = 0 then
        if s =0 then
            return 1;
        else
            return 0 ;
        end if;
    else
        add( procname(n-1,k,s-c*n^2),c=0..k) ;
    end if;
end proc:
A183953 := proc(n,k)
    A183953rec(n,k,k*n^2) ;
end proc:
seq(seq( A183953(n,d-n),n=1..d-1),d=2..12) ; # R. J. Mathar, Mar 08 2021

r[n_, k_, s_] := r[n, k, s] = Which[s < 0, 0, n == 0, If[s == 0, 1, 0], True, Sum[r[n-1, k, s-c*n^2], {c, 0, k}]];
T[n_, k_] := r[n, k, k*n^2];
Table[Table[T[n, d-n], {n, 1, d-1}], {d, 2, 14}] // Flatten (* Jean-François Alcover, Jul 22 2022, after R. J. Mathar *)

A184240 T(n,k)=Number of strings of numbers x(i=1..n) in 0..k with sum i^2x(i)^2 equal to n^2k^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 5, 4, 2, 1, 1, 2, 2, 3, 5, 4, 1, 1, 1, 1, 2, 15, 15, 41, 7, 3, 1, 1, 1, 2, 11, 24, 31, 36, 19, 3, 1, 1, 3, 1, 30, 43, 213, 119, 199, 47, 4, 1, 1, 2, 5, 5, 47, 176, 356, 555, 420, 71, 2, 1, 1, 2, 3, 79, 88, 706, 962, 2531, 2138

Offset: 1

R. H. Hardin Jan 10 2011

Table starts
.1..1...1....1....1.....1.....1......1......1.......1.......1.......1.......1
.1..1...1....1....1.....1.....1......1......1.......1.......1.......1.......2
.1..1...2....1....1.....2.....1......1......3.......2.......2.......2.......2
.1..1...1....1....2.....2.....2......1......5.......3.......3.......2.......4
.2..2...5....3...15....11....30......5.....79......30......99......30.....129
.1..4...5...15...24....43....47.....88....155.....251.....249.....349.....507
.2..4..41...31..213...176...706....571...2355....1508....5431....3445...10802
.1..7..36..119..356...962..2108...3719...6688...13801...22721...31805...53339
.3.19.199..555.2531..5920.19178..32608..85718..133989..284012..450961..850956
.3.47.420.2138.9710.32442.97714.229978.570177.1186420.2444060.4476795.8278231

			Some solutions for n=6 k=5
..4....2....0....0....0....4....0....4....4....0....0....0....0....0....0....0
..2....4....2....5....4....5....4....0....4....2....5....5....3....4....0....4
..2....0....4....0....4....4....4....4....4....4....4....0....4....0....0....2
..4....4....1....4....1....2....4....2....0....2....4....5....3....5....0....5
..0....0....4....4....2....0....4....2....2....2....4....4....0....4....0....4
..4....4....3....2....4....4....1....4....4....4....0....0....4....1....5....0

R. H. Hardin, Table of n, a(n) for n = 1..393

Column 1 is A030273

A184318 T(n,k) = number of strings of numbers x(i=1..n) in 0..k with sum i^2x(i)^3 equal to n^2k^3.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 3, 5, 6, 8, 3, 1, 1, 1, 1, 3, 5, 16, 30, 12, 3, 1, 1, 1, 2, 3, 11, 38, 68, 69, 21, 4, 1, 1, 1, 1, 2, 7, 50, 126, 244, 230, 47, 2, 1, 1, 1, 2, 3, 18, 69, 319, 763, 1106, 662, 117, 7, 1, 1

Offset: 1

R. H. Hardin, Jan 10 2011

Table starts
.1..1...1....1....1....1.....1.....1......1......1......1......1.......1
.1..1...1....1....1....1.....1.....1......1......1......1......1.......1
.1..1...1....1....1....1.....1.....1......1......1......1......1.......1
.1..2...1....2....1....2.....1.....2......1......2......1......2.......1
.2..2...2....2....3....3.....3.....2......3......4......4......4.......2
.1..2...2....5....5...11.....7....18.....15.....26.....37.....47......50
.2..3...6...16...38...50....69...139....188....284....336....550.....608
.1..8..30...68..126..319...574..1056...1637...2680...3831...6128....8009
.3.12..69..244..763.1602..3336..7826..12904..23558..36719..62495...93074
.3.21.230.1106.3133.9343.23562.54515.104508.205011.373576.655153.1053794

			All solutions for n=6 k=5
..2....0....0....0....0
..5....1....0....3....2
..2....4....0....3....3
..1....1....0....4....2
..4....4....0....5....5
..4....4....5....0....3

R. H. Hardin, Table of n, a(n) for n = 1..264

Column 1 is A030273.

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)).

A298857 Number of partitions of the n-th tetrahedral number into distinct tetrahedral numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 5, 5, 10, 12, 17, 15, 22, 30, 56, 65, 72, 92, 172, 219, 299, 368, 478, 810, 1055, 1508, 1778, 2277, 3815, 5214, 7103, 8615, 11614, 18079, 24428, 33704, 42877, 56639, 85597, 116984, 159179, 199356, 268965, 400612, 545674, 740356, 950897, 1261597, 1842307

Offset: 0

Ilya Gutkovskiy, Jan 27 2018

nonn

			a(5) = 2 because fifth tetrahedral number is 35 and we have [35] and [20, 10, 4, 1].

Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers
Index entries for related partition-counting sequences

Cf. A000292, A030273, A279278, A288126, A298269.

Table[SeriesCoefficient[Product[1 + x^(k (k + 1) (k + 2)/6), {k, 1, n}], {x, 0, n (n + 1) (n + 2)/6}], {n, 0, 53}]

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

a(n) = A279278(A000292(n)).

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.

A298934 Number of partitions of n^2 into distinct cubes.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 3, 3, 1, 0, 3, 0, 2, 4, 0, 0, 1, 0, 0, 2, 3, 1, 1, 0, 6, 3, 6, 1, 6, 0, 3, 9, 0, 6, 6, 7, 0, 10, 3, 3, 6, 0, 8, 6, 13, 2, 10, 9, 10, 19, 2, 14, 21, 7, 2, 25

Offset: 0

Ilya Gutkovskiy, Jan 29 2018

nonn

			a(15) = 2 because we have [216, 8, 1] and [125, 64, 27, 8, 1].

Cf. A000290, A000578, A030272, A030273, A218495, A259792, A279329, A298672, A298848, A298935.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(n>i^2*(i+1)^2/4, 0, b(n, i-1)+
      `if`(i^3>n, 0, b(n-i^3, i-1))))
    end:
a:= n-> b(n^2, n):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 29 2018

Table[SeriesCoefficient[Product[1 + x^k^3, {k, 1, Floor[n^(2/3) + 1]}], {x, 0, n^2}], {n, 0, 84}]

a(n) = [x^(n^2)] Product_{k>=1} (1 + x^(k^3)).

a(n) = A279329(A000290(n)).

A298935 Number of partitions of n^3 into distinct squares.

Original entry on oeis.org

1, 1, 0, 0, 1, 5, 8, 40, 96, 297, 1269, 3456, 12839, 46691, 153111, 577167, 2054576, 7602937, 29000337, 110645967, 418889453, 1580667760, 6058528796, 23121913246, 89793473393, 350029321425, 1359919742613, 5340642744919, 20948242218543, 82505892314268

Offset: 0

Ilya Gutkovskiy, Jan 29 2018

nonn

			a(5) = 5 because we have [121, 4], [100, 25], [100, 16, 9], [64, 36, 25] and [64, 36, 16, 9].

Cf. A000290, A000578, A030272, A030273, A033461, A037444, A218494, A298330, A298642, A298934.

Table[SeriesCoefficient[Product[1 + x^k^2, {k, 1, Floor[n^(3/2) + 1]}], {x, 0, n^3}], {n, 0, 29}]

a(n) = [x^(n^3)] Product_{k>=1} (1 + x^(k^2)).

a(n) = A033461(A000578(n)).

A298642 Number of partitions of n^2 into distinct squares > 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 5, 2, 10, 4, 12, 12, 11, 19, 23, 43, 50, 55, 78, 120, 126, 234, 207, 407, 385, 701, 712, 1090, 1231, 1850, 2102, 3054, 3385, 4988, 5584, 7985, 9746, 12205, 15737, 18968, 25157, 30927, 39043, 47708, 61915, 74592, 99554

Offset: 0

Ilya Gutkovskiy, Jan 24 2018

nonn

			a(5) = 2 because we have [25] and [16, 9].

Cf. A000290, A001156, A030273, A033461, A037444, A078134, A092362, A093115, A093116, A280129, A298640.

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

a(n) = A280129(A000290(n)).

cp's OEIS Frontend

A037444 Number of partitions of n^2 into squares.

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A183953 T(n,k) is the number of strings of numbers x(i=1..n) in 0..k with sum i^2*x(i) equal to k*n^2.

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A184240 T(n,k)=Number of strings of numbers x(i=1..n) in 0..k with sum i^2*x(i)^2 equal to n^2*k^2.

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A184318 T(n,k) = number of strings of numbers x(i=1..n) in 0..k with sum i^2*x(i)^3 equal to n^2*k^3.

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

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Maple

Mathematica

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A298857 Number of partitions of the n-th tetrahedral number into distinct tetrahedral numbers.

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

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A298934 Number of partitions of n^2 into distinct cubes.

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Maple

Mathematica

Formula

A183953 T(n,k) is the number of strings of numbers x(i=1..n) in 0..k with sum i^2x(i) equal to kn^2.

A184240 T(n,k)=Number of strings of numbers x(i=1..n) in 0..k with sum i^2x(i)^2 equal to n^2k^2.

A184318 T(n,k) = number of strings of numbers x(i=1..n) in 0..k with sum i^2x(i)^3 equal to n^2k^3.