cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A347808 Expansion of (theta_3(x) - 1)^5 / (16 * (3 - theta_3(x))).

Original entry on oeis.org

1, 1, 1, 6, 7, 8, 19, 25, 37, 56, 76, 122, 170, 233, 347, 494, 700, 991, 1415, 2021, 2855, 4054, 5751, 8164, 11585, 16406, 23285, 33032, 46814, 66375, 94119, 133445, 189193, 268231, 380287, 539184, 764422, 1083722, 1536479, 2178349, 3088333, 4378472, 6207557
Offset: 5

Views

Author

Ilya Gutkovskiy, Sep 14 2021

Keywords

Comments

Number of compositions (ordered partitions) of n into 5 or more squares.

Links

Crossrefs

Cf. A000290, A006456, A337165, A347805, A347806, A347807, A347809.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, `if`(t=0, 1, 0), add((
     s->`if`(s>n, 0, b(n-s, max(0, t-1))))(j^2), j=1..isqrt(n)))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..47);  # Alois P. Heinz, Sep 14 2021
  • Mathematica
    nmax = 47; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^5/(16 (3 - EllipticTheta[3, 0, x])), {x, 0, nmax}], x] // Drop[#, 5] &

Formula

a(n) = Sum_{k=5..n} A337165(n,k). - Alois P. Heinz, Sep 14 2021

A347809 Expansion of (theta_3(x) - 1)^6 / (32 * (3 - theta_3(x))).

Original entry on oeis.org

1, 1, 1, 7, 8, 9, 25, 32, 46, 76, 102, 165, 233, 317, 488, 690, 971, 1395, 1991, 2850, 4024, 5721, 8144, 11550, 16396, 23225, 32987, 46814, 66315, 94069, 133415, 189148, 268181, 380227, 539114, 764387, 1083692, 1536369, 2178299, 3088302, 4378362, 6207477
Offset: 6

Views

Author

Ilya Gutkovskiy, Sep 14 2021

Keywords

Comments

Number of compositions (ordered partitions) of n into 6 or more squares.

Links

Crossrefs

Cf. A000290, A006456, A337165, A347805, A347806, A347807, A347808.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, `if`(t=0, 1, 0), add((
     s->`if`(s>n, 0, b(n-s, max(0, t-1))))(j^2), j=1..isqrt(n)))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..47);  # Alois P. Heinz, Sep 14 2021
  • Mathematica
    nmax = 47; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^6/(32 (3 - EllipticTheta[3, 0, x])), {x, 0, nmax}], x] // Drop[#, 6] &

Formula

a(n) = Sum_{k=6..n} A337165(n,k). - Alois P. Heinz, Sep 14 2021

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

Views

Author

Paul D. Hanna, Apr 12 2013

Keywords

Comments

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

Examples

			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 +...

Crossrefs

Cf. A224608, A219331, A006456, A224366.

Programs

  • PARI
    {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),", "))

Formula

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

Views

Author

Paul D. Hanna, Apr 12 2013

Keywords

Comments

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

Examples

			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 +...

Crossrefs

Cf. A224607, A219331, A006456, A224366.

Programs

  • PARI
    {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),", "))

Formula

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

A294105 Number of compositions (ordered partitions) of n into squares dividing n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 7, 2, 1, 1, 26, 1, 1, 1, 96, 1, 12, 1, 345, 1, 1, 1, 1252, 2, 1, 76, 4544, 1, 1, 1, 17473, 1, 1, 1, 127654, 1, 1, 1, 217286, 1, 1, 1, 788674, 2490, 1, 1, 3182706, 2, 28, 1, 10390321, 1, 14128, 1, 37713313, 1, 1, 1, 136886433, 1, 1, 80396, 579739960, 1, 1, 1, 1803399103, 1, 1
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 28 2017

Keywords

Examples

			a(8) = 7 because 8 has 4 divisors {1, 2, 4, 8} among which 2 are squares {1, 4} therefore we have [4, 4], [4, 1, 1, 1, 1], [1, 4, 1, 1, 1], [1, 1, 4, 1, 1], [1, 1, 1, 4, 1], [1, 1, 1, 1, 4] and [1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A006456, A046951, A100346, A284345.

Programs

  • Maple
    a:= proc(n) option remember; local b, l;
      l, b:= select(issqr, numtheory[divisors](n)),
      proc(m) option remember; `if`(m=0, 1,
         add(`if`(j>m, 0, b(m-j)), j=l))
      end; b(n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 30 2017
  • Mathematica
    Table[SeriesCoefficient[1/(1 - Sum[Boole[Mod[n, k] == 0 && IntegerQ[k^(1/2)]] x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 70}]

A298938 Number of ordered ways of writing n^3 as a sum of n squares of nonnegative integers.

Original entry on oeis.org

1, 1, 1, 4, 5, 686, 13942, 455988, 13617853, 454222894, 18323165948, 802161109047, 42149084452070, 2481730049781672, 157265294178424356, 10977302934685469078, 812821237985857557677, 64539935903231450294134, 5504599828399250884049308
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2018

Keywords

Examples

			a(4) = 5 because we have [64, 0, 0, 0], [16, 16, 16, 16], [0, 64, 0, 0], [0, 0, 64, 0] and [0, 0, 0, 64].

Links

Crossrefs

Cf. A000290, A000578, A006456, A030273, A037444, A066535, A218494, A232173, A287617, A298329, A298330, A298935, A298936, A298939.

Programs

  • Mathematica
    Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^3}], {n, 0, 18}]

Formula

a(n) = [x^(n^3)] (Sum_{k>=0} x^(k^2))^n.

A298939 Number of ordered ways of writing n^3 as a sum of n squares of positive integers.

Original entry on oeis.org

1, 1, 1, 4, 1, 286, 7582, 202028, 6473625, 226029577, 8338249868, 391526193477, 19990594900630, 1159906506684446, 74890158861242740, 5119732406649036418, 380146984328280974281, 30198665638519565614034, 2555354508318427693497565
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2018

Keywords

Examples

			a(3) = 4 because we have [25, 1, 1], [9, 9, 9], [1, 25, 1] and [1, 1, 25].

Links

Crossrefs

Cf. A000290, A000578, A006456, A030273, A037444, A066535, A218494, A232173, A287617, A298329, A298330, A298935, A298937, A298938.

Programs

  • Mathematica
    Table[SeriesCoefficient[(-1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^3}], {n, 0, 18}]

Formula

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

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

Views

Author

Ilya Gutkovskiy, Jan 30 2020

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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

Extensions

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

A339418 Number of compositions (ordered partitions) of n into an even number of squares.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 4, 2, 6, 9, 8, 20, 16, 35, 44, 55, 102, 105, 196, 242, 344, 540, 652, 1084, 1380, 2037, 2964, 3912, 6042, 7976, 11776, 16634, 22968, 33963, 46156, 67457, 94510, 133180, 192316, 266514, 385338, 540138, 767008, 1094576, 1534704, 2200821, 3094248
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2020

Keywords

Examples

			a(9) = 6 because we have [4, 1, 1, 1, 1, 1], [1, 4, 1, 1, 1, 1], [1, 1, 4, 1, 1, 1], [1, 1, 1, 4, 1, 1], [1, 1, 1, 1, 4, 1] and [1, 1, 1, 1, 1, 4].

Links

Crossrefs

Cf. A000290, A006456, A034008, A317665, A339364, A339416, A339419.

Programs

  • Maple
    b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+2*g-1, g+1 od; r fi
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 03 2020
  • Mathematica
    nmax = 47; CoefficientList[Series[4/(3 + 2 EllipticTheta[3, 0, x] - EllipticTheta[3, 0, x]^2), {x, 0, nmax}], x]

Formula

G.f.: 4 / (3 + 2 * theta_3(x) - theta_3(x)^2), where theta_3() is the Jacobi theta function.
a(n) = (A006456(n) + A317665(n)) / 2.
a(n) = Sum_{k=0..n} A006456(k) * A317665(n-k).

A347710 Number of compositions (ordered partitions) of n into at most 3 squares.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 0, 1, 4, 2, 3, 1, 2, 6, 0, 1, 5, 4, 3, 2, 6, 3, 0, 3, 3, 8, 4, 0, 8, 6, 0, 1, 6, 5, 6, 4, 2, 9, 0, 2, 11, 6, 3, 3, 8, 6, 0, 1, 7, 9, 6, 2, 8, 12, 0, 6, 6, 2, 9, 0, 8, 12, 0, 1, 10, 12, 3, 5, 12, 6, 0, 4, 5, 14, 7, 3, 12, 6, 0, 2
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2021

Keywords

Links

Crossrefs

Cf. A000290, A002654, A006456, A063691, A337165, A347711, A347712, A347713.

Formula

a(n) = Sum_{k=0..3} A337165(n,k). - Alois P. Heinz, Sep 10 2021
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