A103198 -id:A103198 - cp's OEIS frontend

A003099 a(n) = Sum_{k=0..n} binomial(n,k^2).

1, 2, 3, 4, 6, 11, 22, 43, 79, 137, 231, 397, 728, 1444, 3018, 6386, 13278, 26725, 51852, 97243, 177671, 320286, 579371, 1071226, 2053626, 4098627, 8451288, 17742649, 37352435, 77926452, 159899767, 321468048, 632531039, 1219295320, 2308910353, 4314168202

Offset: 0

N. J. A. Sloane, Henry W. Gould

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

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Henry W. Gould, Fibonomial Catalan numbers: arithmetic properties and a table of the first fifty numbers, Abstract 71T-A216, Notices Amer. Math. Soc, 1971, page 938. [Annotated scanned copy of abstract]
Henry W. Gould, Letter to N. J. A. Sloane, Nov 1973, and various attachments.
Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.

Cf. A010052, A206849.

Partial sums of A103198.

[(&+[Binomial(n, j^2): j in [0..n]]): n in [0..50]]; // G. C. Greubel, Oct 26 2022

Table[Sum[Binomial[n, k^2], {k, 0, Sqrt[n]}], {n, 0, 50}] (* T. D. Noe, Sep 10 2011 *)

a(n)=sum(k=0,sqrtint(n),binomial(n,k^2)) \\ Charles R Greathouse IV, Mar 26 2013

def A003099(n): return sum( binomial(n,k^2) for k in range(isqrt(n)+1))
[A003099(n) for n in range(50)] # G. C. Greubel, Oct 26 2022

a(n)*sqrt(n)/2^n is bounded: lim sup a(n)*sqrt(n)/2^n = 0.82... and lim inf a(n)*sqrt(n)/2^n = 0.58... - Benoit Cloitre, Nov 14 2003 [These constants are sqrt(2/Pi) * JacobiTheta3(0,exp(-4)) = 0.827112271364145742... and sqrt(2/Pi) * JacobiTheta2(0,exp(-4)) = 0.587247586271786487... - Vaclav Kotesovec, Jan 15 2023]
Binomial transform of the characteristic function of squares A010052. - Carl Najafi, Sep 09 2011
G.f.: (1/(1 - x)) * Sum_{k>=0} (x/(1 - x))^(k^2). - Ilya Gutkovskiy, Jan 22 2024

More terms from Carl Najafi, Sep 09 2011

A323525 Number of ways to arrange the parts of a multiset whose multiplicities are the prime indices of n into a square matrix.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 0, 6, 4, 0, 12, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 84, 0, 0, 72, 0, 0, 0, 0, 0, 0, 0, 0, 126, 252, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jan 17 2019

nonn

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The a(9) = 6 matrices:
  [1 1] [1 2] [1 2] [2 1] [2 1] [2 2]
  [2 2] [1 2] [2 1] [1 2] [2 1] [1 1]
The a(38) = 9 matrices:
  [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 1]
  [1 1 1] [1 1 1] [1 1 1] [1 1 2] [1 2 1] [2 1 1] [1 1 1] [1 1 1] [1 1 1]
  [1 1 2] [1 2 1] [2 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1]

The positions of 0's are numbers whose sum of prime indices is not a perfect square (A323527).
The positions of 1's are primes indexed by squares (A323526).
Cf. A008480, A026478, A056239, A103198, A112798, A120732, A318284, A318762, A323519, A323528, A323531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,Reverse[primeMS[n]]];
Table[If[IntegerQ[Sqrt[Total[primeMS[n]]]],Length[Permutations[nrmptn[n]]],0],{n,100}]

If A056239(n) is a perfect square, a(n) = A318762(n). Otherwise, a(n) = 0.

A323519 a(n) is the number of ways to fill a square matrix with the multiset of prime factors of n, if the number of prime factors (counted with multiplicity) is a perfect square, and a(n) = 0 otherwise.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 6, 1, 0, 0, 4, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 4, 0, 4, 0, 0, 1, 12, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 12, 0, 0

Offset: 1

Gus Wiseman, Jan 17 2019

nonn

			The a(60) = 12 matrices:
  [2 2] [2 2] [2 3] [2 3] [2 5] [2 5] [3 2] [3 2] [3 5] [5 2] [5 2] [5 3]
  [3 5] [5 3] [2 5] [5 2] [2 3] [3 2] [2 5] [5 2] [2 2] [2 3] [3 2] [2 2]

Positions of 0's are A323521.
Positions of 1's are A323520.
Cf. A000290, A008480, A026478, A056239, A089299, A103198, A112798, A120732, A323433, A323525, A323531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[IntegerQ[Sqrt[PrimeOmega[n]]],Length[Permutations[primeMS[n]]],0],{n,100}]

If A001222(n) is a perfect square, then a(n) = A008480(n). Otherwise, a(n) = 0.

A323522 Number of ways to fill a square matrix with the parts of a strict integer partition of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 25, 49, 73, 121, 145, 217, 265, 361, 433, 553, 649, 817, 937, 1129, 1297, 1537, 1729, 2017, 2257, 2593, 2881, 3265, 3601, 4057, 4441, 4945, 5401, 5977, 6481, 7129, 7705, 8425, 9073, 9865, 373465, 374353, 738025, 1101865, 1828513

Offset: 0

Gus Wiseman, Jan 17 2019

nonn

			The a(10) = 25 matrices:
  [10]
.
  [4 3] [4 3] [4 2] [4 2] [4 1] [4 1] [3 4] [3 4]
  [2 1] [1 2] [3 1] [1 3] [3 2] [2 3] [2 1] [1 2]
.
  [3 2] [3 2] [3 1] [3 1] [2 4] [2 4] [2 3] [2 3]
  [4 1] [1 4] [4 2] [2 4] [3 1] [1 3] [4 1] [1 4]
.
  [2 1] [2 1] [1 4] [1 4] [1 3] [1 3] [1 2] [1 2]
  [4 3] [3 4] [3 2] [2 3] [4 2] [2 4] [4 3] [3 4]

Alois P. Heinz, Table of n, a(n) for n = 0..7000

Cf. A000009, A089299, A103198 (non-strict case), A120732, A323431, A323434, A323519, A323523, A323525, A323529.

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
      -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
    end:
a:= n-> (l-> add(l[i^2+1]*(i^2)!, i=0..floor(sqrt(nops(l)-1))))(b(n$2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 17 2019

Table[Sum[(k^2)!*Length[Select[IntegerPartitions[n,{k^2}],UnsameQ@@#&]],{k,n}],{n,20}]
(* Second program: *)
q[n_, k_] := q[n, k] = If[n < k || k < 1, 0,
     If[n == 1, 1, q[n-k, k] + q[n-k, k-1]]];
a[n_] := If[n == 0, 1, Sum[(k^2)! q[n, k^2], {k, 0, n}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 20 2021 *)

a(n) = Sum_{k >= 0} (k^2)! * Q(n, k^2) where Q = A008289.

A280351 Expansion of Sum_{k>=0} (x/(1 - x))^(k^3).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1717, 3433, 6436, 11441, 19449, 31825, 50389, 77521, 116281, 170545, 245158, 346105, 480701, 657802, 888058, 1184419, 1564435, 2063206, 2799487, 4272049, 8544097, 23535821, 77331981, 262534537, 865287625, 2720095405

Offset: 0

Ilya Gutkovskiy, Jan 01 2017

Number of compositions of n into a cube number of parts.

			a(9) = 9 because we have:
[1]  [9]
[2]  [2, 1, 1, 1, 1, 1, 1, 1]
[3]  [1, 2, 1, 1, 1, 1, 1, 1]
[4]  [1, 1, 2, 1, 1, 1, 1, 1]
[5]  [1, 1, 1, 2, 1, 1, 1, 1]
[6]  [1, 1, 1, 1, 2, 1, 1, 1]
[7]  [1, 1, 1, 1, 1, 2, 1, 1]
[8]  [1, 1, 1, 1, 1, 1, 2, 1]
[9]  [1, 1, 1, 1, 1, 1, 1, 2]

Index entries for sequences related to compositions

Cf. A000578, A052467, A103198, A280352.

a := n -> ifelse(n = 0, 1, add(binomial(n - 1, k^3 - 1), k = 1..floor(n^(1/3)))):
seq(a(n), n = 0..39); # Peter Luschny, Dec 23 2022

nmax = 39; CoefficientList[Series[Sum[(x/(1 - x))^k^3, {k, 0, nmax}], {x, 0, nmax}], x]

a(0) = 1; a(n) = Sum_{k=1..floor(n^(1/3))} binomial(n-1, k^3-1) for n > 0. - Jerzy R Borysowicz, Dec 22 2022

a(0)=1 prepended by Alois P. Heinz, Dec 17 2022

A280352 Expansion of Sum_{k>=1} (x/(1 - x))^(k*(k+1)/2).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 22, 43, 85, 164, 308, 573, 1079, 2081, 4097, 8129, 16049, 31315, 60402, 115806, 222416, 430791, 843987, 1670054, 3322167, 6606936, 13078586, 25714238, 50230292, 97708338, 189921842, 370216757, 725680489, 1431888173, 2842060970, 5662371069

Offset: 1

Ilya Gutkovskiy, Jan 01 2017

Number of compositions of n into a triangular number of parts.

			a(5) = 7 because we have:
  [1]  [5]
  [2]  [3, 1, 1]
  [3]  [1, 3, 1]
  [4]  [1, 1, 3]
  [5]  [2, 2, 1]
  [6]  [2, 1, 2]
  [7]  [1, 2, 2]

Vaclav Kotesovec, Plot of a(n) / (2^(n-1)/sqrt(n)) for n = 1..10000
Index entries for sequences related to compositions

Cf. A000217, A052467, A103198, A280351.

nmax = 36; Rest[CoefficientList[Series[Sum[(x/(1 - x))^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 40; Rest[CoefficientList[Series[-1 + EllipticTheta[2, 0, Sqrt[x/(1-x)]]/(2*(x/(1-x))^(1/8)), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 01 2017 *)

a(n) = sum(k=1, (sqrtint(8*n+1)-1)\2, binomial(n-1, k*(k+1)/2-1)) \\ Andrew Howroyd, Jan 14 2023

G.f.: Sum_{k>=1} (x/(1-x))^(k*(k+1)/2).
a(n) = Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} binomial(n-1, k*(k+1)/2-1). - Jerzy R Borysowicz, Dec 26 2022
Conjecture: a(n+1)/a(n) ~ 2. - Jerzy R Borysowicz, Jan 14 2023
Conjecture: abs(b(n)-1) < 0.015, where b(n) = a(n)*sqrt(n)/2^(n-1), for n > 781; b(n) does not have a limit. - Jerzy R Borysowicz, Feb 17 2023

A323523 Number of positive integer square matrices with entries summing to n and equal row and column sums.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 12, 1, 7, 22, 9, 1, 64, 1, 34, 121, 11, 1, 525, 2, 13, 407, 2022, 1, 801, 1, 10163, 1036, 17, 6211, 41735, 1, 19, 2212, 285784, 1, 3822, 1, 381446, 2229142, 23, 1, 1189540, 2, 22069276, 7261, 2309410, 1, 20943183, 164176641

Offset: 0

Gus Wiseman, Jan 17 2019

nonn

Also the number of non-normal semi-magic squares with positive integer entries summing to n.

			The a(12) = 12 matrices:
  [12]
.
  [1 5] [5 1] [2 4] [4 2] [3 3]
  [5 1] [1 5] [4 2] [2 4] [3 3]
.
  [1 1 2] [1 1 2] [1 2 1] [1 2 1] [2 1 1] [2 1 1]
  [1 2 1] [2 1 1] [1 1 2] [2 1 1] [1 1 2] [1 2 1]
  [2 1 1] [1 2 1] [2 1 1] [1 1 2] [1 2 1] [1 1 2]

Andrew Howroyd, Table of n, a(n) for n = 0..200 (terms 0..59 from Chai Wah Wu)

Cf. A000290, A006052, A007016, A103198, A120732, A257493, A321719, A321722, A323349, A323523, A323524, A323529.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnsqrs[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),And[SameQ@@Length/@#,Length[#]==0||Length[#]==Length[First[#]]]&];
Table[Sum[Length[Select[ptnsqrs[Times@@Prime/@y],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]],{y,IntegerPartitions[n]}],{n,10}]

a(p) = 1 and a(p^2) = 2 for p prime (see comment in A323349). - Chai Wah Wu, Jan 20 2019

a(n) = Sum_{d|n, d<=n/d} A257493(d, n/d-d) for n > 0. - Andrew Howroyd, Apr 10 2020

a(16)-a(55) from Chai Wah Wu, Jan 20 2019

A307522 Expansion of Product_{k>=1} ((1 + x)^k - x^k)/((1 + x)^k + x^k).

Original entry on oeis.org

1, -2, 2, -2, 4, -10, 22, -42, 72, -116, 188, -332, 662, -1432, 3148, -6736, 13784, -26894, 50254, -90782, 160856, -285230, 518170, -983710, 1964800, -4090002, 8705322, -18582722, 39219572, -81148034, 163946630, -323136562, 622125982, -1173528562, 2179230066

Offset: 0

Seiichi Manyama, Apr 12 2019

Cf. A002448, A103198, A307520, A307521, A318570.

a(n) := 2*(-1)^n*add( binomial(n-1, n-k^2), k = 1..floor(sqrt(n))):
print(1, seq(a(n), n = 1..40)); # Peter Bala, Dec 31 2024

m = 34; CoefficientList[Series[Product[((1 + x)^k - x^k)/((1 + x)^k + x^k), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)

my(N=66, x='x+O('x^N)); Vec(prod(k=1, N, ((1+x)^k-x^k)/((1+x)^k+x^k)))

G.f.: theta_4(x/(1 + x)), where theta_4() is the Jacobi theta function.
From Peter Bala, Dec 31 2024: (Start)
For n >= 1, a(n) = 2 * (-1)^n * Sum_{k = 1..floor(sqrt(n))} binomial(n-1, n-k^2).
For n >= 1, |a(n)| = 2 * A103198(n). (End)

A339445 Number of partitions of n into squares such that the number of parts is a square.

Original entry on oeis.org

1, 1, 0, 0, 2, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 2, 3, 1, 2, 2, 2, 2, 2, 2, 3, 5, 2, 4, 6, 1, 4, 6, 3, 7, 6, 4, 10, 6, 4, 10, 9, 6, 11, 10, 8, 10, 10, 11, 14, 16, 11, 15, 19, 10, 17, 22, 13, 24, 23, 16, 28, 21, 18, 33, 30, 24, 33, 33, 29, 33, 37, 33, 43, 45, 35, 49

Offset: 0

Ilya Gutkovskiy, Dec 05 2020

nonn

			                                    [1 1 1]
                          [1 4]     [1 1 1]
a(23) = 2 because we have [9 9] and [4 4 9].

Robert Israel, Table of n, a(n) for n = 0..2500
Eric Weisstein's World of Mathematics, Square Number
Index entries for sequences related to partitions
Index entries for sequences related to sums of squares

Cf. A000290, A001156, A085755, A089333, A103198, A323522, A339444.

g:= proc(n, k, m)
  # number of partitions of n into k parts which are squares > m^2
   option remember; local r;
  if k = 0 then if n = 0 then return 1 else return 0 fi fi;
  if n < k*(m+1)^2 then return 0 fi;
  add(procname(n-r*(m+1)^2, k-r, m+1), r =max(0, ceil((k*(m+2)^2-n)/(2*m+3))) .. k)
end proc:
f:= proc(n) local k; add(g(n,k^2,0),k=1..floor(sqrt(n))) end proc:
f(0):= 1:
map(f, [$0..100]); # Robert Israel, Oct 26 2023

A323524 Number of integer partitions of n whose parts can be arranged into a square matrix with equal row and column sums.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 5, 1, 4, 4, 6, 1, 10, 1, 7, 10, 6, 1, 24, 2, 7, 22, 18, 1, 38, 1, 35, 43, 9, 6, 124, 1, 10, 77, 158, 1, 110, 1, 285, 186, 12, 1, 742, 2, 170, 203, 1110, 1, 285, 480, 2115, 306, 15, 1

Offset: 0

Gus Wiseman, Jan 17 2019

			The a(12) = 5 integer partitions are (12), (5,5,1,1), (4,4,2,2), (3,3,3,3), (2,2,2,1,1,1,1,1,1). For example, such a matrix for (2,2,2,1,1,1,1,1,1) is:
  [1 1 2]
  [2 1 1]
  [1 2 1]

Cf. A006052, A007016, A103198, A120732, A321719, A321722, A321725, A323306, A323347, A323349, A323523.

a(p) = 1 and a(p^2) = 2 for p prime (see comment in A323349). - Chai Wah Wu, Jan 20 2019

a(16)-a(59) from Chai Wah Wu, Jan 20 2019

cp's OEIS Frontend

A003099 a(n) = Sum_{k=0..n} binomial(n,k^2).

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A323525 Number of ways to arrange the parts of a multiset whose multiplicities are the prime indices of n into a square matrix.

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A323519 a(n) is the number of ways to fill a square matrix with the multiset of prime factors of n, if the number of prime factors (counted with multiplicity) is a perfect square, and a(n) = 0 otherwise.

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A323522 Number of ways to fill a square matrix with the parts of a strict integer partition of n.

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A280351 Expansion of Sum_{k>=0} (x/(1 - x))^(k^3).

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A280352 Expansion of Sum_{k>=1} (x/(1 - x))^(k*(k+1)/2).

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A323523 Number of positive integer square matrices with entries summing to n and equal row and column sums.

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