A224366 -id:A224366 - cp's OEIS frontend

A290247 Number of compositions (ordered partitions) of n^3 into cubes.

1, 1, 2, 120, 290250, 107320441096, 21715974961480054078, 8487986089807555456140271121440, 22615863021403796876556069287242400147213424924, 1449638083412288206280215383952017948209203861522683138464747658192

Offset: 0

Ilya Gutkovskiy, Jul 24 2017

nonn

			a(2) = 2 because 2^3 = 8 and we have [8], [1, 1, 1, 1, 1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..22
Index entries for sequences related to compositions

Cf. A000578, A023358, A030272, A224366, A259792.

b:= proc(n) option remember; local i; if n=0 then 1
      else 0; for i while i^3<=n do %+b(n-i^3) od fi
    end:
a:= n-> b(n^3):
seq(a(n), n=0..10);  # Alois P. Heinz, Aug 12 2017

Table[SeriesCoefficient[1/(1 - Sum[x^k^3, {k, 1, n}]), {x, 0, n^3}], {n, 0, 9}]

a(n) = [x^(n^3)] 1/(1 - Sum_{k>=1} x^(k^3)).

a(n) = A023358(A000578(n)).

A224677 Number of compositions of n*(n+1)/2 into sums of positive triangular numbers.

Original entry on oeis.org

1, 1, 2, 7, 40, 351, 4876, 104748, 3487153, 179921982, 14387581923, 1783124902639, 342504341570010, 101962565961894431, 47044167891731682278, 33640402686770010577421, 37282664267078280296013183, 64038780633654058635677191329, 170478465430659361252118580217675

Offset: 0

Paul D. Hanna, Apr 14 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..102

Cf. A023361, A224366, A224679.

b:= proc(n) option remember; local i; if n=0 then 1 else 0;
      for i while i*(i+1)/2<=n do %+b(n-i*(i+1)/2) od; %  fi
    end:
a:= n-> b(n*(n+1)/2):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 05 2018

b[n_] := b[n] = If[n==0, 1, Sum[If[IntegerQ[Sqrt[8j+1]], b[n-j], 0], {j, 1, n}]];
a[n_] := b[n(n+1)/2];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 31 2020, after Alois P. Heinz in A023361 *)

{a(n)=polcoeff(1/(1-sum(r=1,n+1, x^(r*(r+1)/2)+x*O(x^(n*(n+1)/2)))), n*(n+1)/2)}
for(n=0, 20, print1(a(n), ", "))

a(n) = A023361(n*(n+1)/2), where A023361(n) is the number of compositions of n into positive triangular numbers.

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

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.

A224679 Number of compositions of n^2 into sums of positive triangular numbers.

Original entry on oeis.org

1, 1, 3, 25, 546, 28136, 3487153, 1038115443, 742336894991, 1275079195875471, 5260826667789867957, 52137661179700350278531, 1241165848412448464485760897, 70972288312605764017275784402928, 9748291749334923037419108242002717050

Offset: 0

Paul D. Hanna, Apr 14 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..72

Cf. A023361, A224366, A224677.

b:= proc(n) option remember; local i; if n=0 then 1 else 0;
      for i while i*(i+1)/2<=n do %+b(n-i*(i+1)/2) od; %  fi
    end:
a:= n-> b(n^2):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 05 2018

b[n_] := b[n] = Module[{i, j = If[n == 0, 1, 0]}, For[i = 1, i(i+1)/2 <= n, i++, j += b[n-i(i+1)/2]]; j];
a[n_] := b[n^2];
a /@ Range[0, 20] (* Jean-François Alcover, Nov 04 2020, after Alois P. Heinz *)

{a(n)=polcoeff(1/(1-sum(r=1,n+1, x^(r*(r+1)/2)+x*O(x^(n^2)))), n^2)}
for(n=0, 20, print1(a(n), ", "))

a(n) = A023361(n^2), where A023361(n) = number of compositions of n into positive triangular numbers.

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

A298640 Number of compositions (ordered partitions) of n^2 into squares > 1.

Original entry on oeis.org

1, 0, 1, 1, 2, 8, 12, 129, 874, 9630, 167001, 3043147, 72844510, 2423789655, 106665874384, 6156805673648, 470151743582651, 47558937432498729, 6363358599941131580, 1126147544855148769425, 263646401550138303553708, 81649922556593759124887197

Offset: 0

Ilya Gutkovskiy, Jan 24 2018

nonn

			a(5) = 8 because we have [25], [16, 9], [9, 16], [9, 4, 4, 4, 4], [4, 9, 4, 4, 4], [4, 4, 9, 4, 4], [4, 4, 4, 9, 4] and [4, 4, 4, 4, 9].

Cf. A000290, A006456, A078134, A224366, A280542, A298642.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j^2), j=2..isqrt(n)))
    end:
a:= n-> b(n^2):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 05 2018

b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j^2], {j, 2, Floor @ Sqrt[n]}]];
a[n_] := b[n^2];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 21 2018, after Alois P. Heinz *)

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

a(n) = A280542(A000290(n)).

A224607 a(n) = A219331(n^2).

Original entry on oeis.org

1, 5, 28, 269, 6181, 286790, 26808447, 5037694829, 1902773895751, 1444565587750055, 2204357811343981558, 6761166975496300074014, 41682712965722542326438411, 516517498759950258411494666787, 12864972023450485679400300069493738

Offset: 1

Paul D. Hanna, Apr 12 2013

nonn

A219331 is the logarithmic derivative of A006456, where A006456(n) is the number of compositions of n into sums of squares.

			L.g.f.: L(x) = x + 5*x^2/2 + 28*x^3/3 + 269*x^4/4 + 6181*x^5/5 + 286790*x^6/6 +...
where
exp(L(x)) = 1 + x + 3*x^2 + 12*x^3 + 81*x^4 + 1335*x^5 + 49309*x^6 + 3882180*x^7 +...+ A224608(n)*x^n +...

Cf. A224608, A219331, A006456, A224366.

{a(n)=n^2*polcoeff(-log(1-sum(r=1,n,x^(r^2)+x*O(x^(n^2)))),n^2)}
for(n=1,30,print1(a(n),", "))

Logarithmic derivative of A224608.

A224608 G.f.: exp( Sum_{n>=1} A219331(n^2)*x^n/n ).

Original entry on oeis.org

1, 1, 3, 12, 81, 1335, 49309, 3882180, 633703214, 212061201327, 144669917959584, 200541263416077021, 563631413420071614333, 3206926569346230863485855, 36897315109526505791310840932, 857701705296285206387609947414980, 40254707002970300021370965171570478599

Offset: 0

Paul D. Hanna, Apr 12 2013

nonn

A219331 is the logarithmic derivative of A006456, where A006456(n) is the number of compositions of n into sums of squares.

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 81*x^4 + 1335*x^5 + 49309*x^6 +...
where
log(A(x)) = x + 5*x^2/2 + 28*x^3/3 + 269*x^4/4 + 6181*x^5/5 + 286790*x^6/6 +...+ A219331(n^2)*x^n/n +...

Cf. A224607, A219331, A006456, A224366.

{A219331(n)=n*polcoeff(-log(1-sum(r=1,sqrtint(n+1),x^(r^2)+x*O(x^n))),n)}
{a(n)=polcoeff(exp(sum(m=1,n,A219331(m^2)*x^m/m)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

Logarithmic derivative yields A224607, where A224607(n) = A219331(n^2).

A331884 Number of compositions (ordered partitions) of n^2 into distinct squares.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 7, 1, 31, 123, 151, 121, 897, 7351, 5415, 14881, 48705, 150583, 468973, 1013163, 1432471, 1730023, 50432107, 14925241, 125269841, 74592537, 241763479, 213156871, 895153173, 7716880623, 2681163865, 3190865761, 22501985413, 116279718801

Offset: 0

Ilya Gutkovskiy, Jan 30 2020

nonn

			a(5) = 3 because we have [25], [16, 9] and [9, 16].

Cf. A000290, A006456, A030273, A032020, A037444, A105152, A224366, A232173, A280129, A298640, A331844.

b:= proc(n, i, p) option remember;
      `if`(i*(i+1)*(2*i+1)/6n, 0, b(n-i^2, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n^2, n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 30 2020

b[n_, i_, p_] := b[n, i, p] = If[i(i+1)(2i+1)/6 < n, 0, If[n == 0, p!, If[i^2 > n, 0, b[n - i^2, i - 1, p + 1]] + b[n, i - 1, p]]];
a[n_] := b[n^2, n, 0];
a /@ Range[0, 35] (* Jean-François Alcover, Nov 08 2020, after Alois P. Heinz *)

a(n) = A331844(A000290(n)).

a(24)-a(34) from Alois P. Heinz, Jan 30 2020

A347590 Number of compositions (ordered partitions) of n^2 into at most n squares.

Original entry on oeis.org

1, 1, 1, 4, 2, 20, 188, 1031, 8777, 62528, 437160, 4185739, 38642386, 383969125, 4149154916, 45160025119, 514181220266, 6133093344169, 75135177511922, 962729735639323, 12745694628358530, 173304634479902187, 2433186864257121180, 35030916937968941062

Offset: 0

Ilya Gutkovskiy, Sep 08 2021

nonn

Index entries for sequences related to sums of squares

Cf. A000290, A006456, A105152, A224366, A298330, A347591, A347592.

A232266 Triangle where T(n,k) = number of compositions of n^2 - k^2 into sums of squares for k=0..n, n>=0, as read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 11, 7, 3, 1, 124, 88, 30, 5, 1, 2870, 2024, 710, 124, 11, 1, 133462, 94137, 33033, 5767, 502, 22, 1, 12477207, 8800750, 3088365, 539192, 46832, 2024, 43, 1, 2344649612, 1653790807, 580347968, 101321507, 8800750, 380315, 8176, 88, 1, 885591183971, 624648802700, 219201637352, 38269865019, 3324109524, 143647802, 3088365, 33033, 175, 1

Offset: 0

Paul D. Hanna, Nov 21 2013

			Triangle begins:
1;
1, 1;
2, 1, 1;
11, 7, 3, 1;
124, 88, 30, 5, 1;
2870, 2024, 710, 124, 11, 1;
133462, 94137, 33033, 5767, 502, 22, 1;
12477207, 8800750, 3088365, 539192, 46832, 2024, 43, 1;
2344649612, 1653790807, 580347968, 101321507, 8800750, 380315, 8176, 88, 1;
885591183971, 624648802700, 219201637352, 38269865019, 3324109524, 143647802, 3088365, 33033, 175, 1; ...
where T(n,k) = coefficient of x^(n^2-k^2) in the series:
1/(1 - x - x^4 - x^9 - x^16 - x^25 - x^36 -...- x^(n^2) -...) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 7*x^8 + 11*x^9 + 16*x^10 + 22*x^11 + 30*x^12 + 43*x^13 + 62*x^14 + 88*x^15 + 124*x^16 + 175*x^17 + 249*x^18 + 354*x^19 + 502*x^20 + 710*x^21 + 1006*x^22 + 1427*x^23 + 2024*x^24 + 2870*x^25 +...

Cf. A006456, A224366.

{T(n,k)=polcoeff(1/(1-sum(m=1,n+1,x^(m^2))+x*O(x^(n^2-k^2))),n^2-k^2)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

T(n,k) = A006456(n^2-k^2).
T(n,k) = [x^(n^2-k^2)] 1/(1 - Sum_{j>=1} x^(j^2)).
T(n,0) = Sum_{k=1..n} T(n,k) for n>=1.

cp's OEIS Frontend

A290247 Number of compositions (ordered partitions) of n^3 into cubes.

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A224677 Number of compositions of n*(n+1)/2 into sums of positive triangular 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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A224679 Number of compositions of n^2 into sums of positive triangular numbers.

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A298640 Number of compositions (ordered partitions) of n^2 into squares > 1.

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A224607 a(n) = A219331(n^2).

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A224608 G.f.: exp( Sum_{n>=1} A219331(n^2)*x^n/n ).

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A331884 Number of compositions (ordered partitions) of n^2 into distinct squares.

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