A352373 -id:A352373 - cp's OEIS frontend

A348410 Number of nonnegative integer solutions to n = Sum_{i=1..n} (a_i + b_i), with b_i even.

1, 1, 5, 19, 85, 376, 1715, 7890, 36693, 171820, 809380, 3830619, 18201235, 86770516, 414836210, 1988138644, 9548771157, 45948159420, 221470766204, 1069091485500, 5167705849460, 25009724705460, 121171296320475, 587662804774890, 2852708925078675, 13859743127937876

Offset: 0

César Eliud Lozada, Oct 17 2021

Suppose n objects are to be distributed into 2n baskets, half of these white and half black. White baskets may contain 0 or any number of objects, while black baskets may contain 0 or an even number of objects. a(n) is the number of distinct possible distributions.

			Some examples (semicolon separates white basket from black baskets):
For n=1: {{1 ; 0}} - Total possible ways: 1.
For n=2: {{0, 0 ; 0, 2}, {0, 0 ; 2, 0}, {0, 2 ; 0, 0}, {1, 1 ; 0, 0}, {2, 0 ; 0, 0}} - Total possible ways: 5.

Alois P. Heinz, Table of n, a(n) for n = 0..1441

Cf. A033715, A033716, A034297, A063020, A088218, A234839, A294860, A348474, A351856, A351857, A352373.

b:= proc(n, t) option remember; `if`(t=0, 1-signum(n),
      add(b(n-j, t-1)*(1+iquo(j, 2)), j=0..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 17 2021

(* giveList=True produces the list of solutions *)
(* giveList=False gives the number of solutions *)
counter[objects_, giveList_: False] :=
  Module[{n = objects, nb, eq1, eqa, eqb, eqs, var, sol, var2, list},
   nb = n;
   eq1 = {Total[Map[a[#] + 2*b[#] &, Range[nb]]] - n == 0};
   eqa = {And @@ Map[0 <= a[#] <= n &, Range[nb]]};
   eqb = {And @@ Map[0 <= b[#] <= n &, Range[nb]]};
   eqs = {And @@ Join[eq1, eqa, eqb]};
   var = Flatten[Map[{a[#], b[#]} &, Range[nb]]];
   var = Join[Map[a[#] &, Range[nb]], Map[b[#] &, Range[nb]]];
   sol = Solve[eqs, var, Integers];
   var2 = Join[Map[a[#] &, Range[nb]], Map[2*b[#] &, Range[nb]]];
   list = Sort[Map[var2 /. # &, sol]];
   list = Map[StringReplace[ToString[#], {"," -> " ;"}, n] &, list];
   list = Map[StringReplace[#, {";" -> ","}, n - 1] &, list];
   Return[
    If[giveList, Print["Total: ", Length[list]]; list, Length[sol]]];
   ];
(* second program: *)
b[n_, t_] := b[n, t] = If[t == 0, 1 - Sign[n], Sum[b[n - j, t - 1]*(1 + Quotient[j, 2]), {j, 0, n}]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 16 2023, after Alois P. Heinz *)

Conjecture: D-finite with recurrence +7168*n*(2*n-1)*(n-1)*a(n) -64*(n-1)*(1759*n^2-5294*n+5112)*a(n-1) +12*(7561*n^3-75690*n^2+165271*n-101070)*a(n-2) +5*(110593*n^3-743946*n^2+1659971*n-1232778)*a(n-3) +2680*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4)=0. - R. J. Mathar, Oct 19 2021
From Vaclav Kotesovec, Nov 01 2021: (Start)
Recurrence (of order 2): 16*(n-1)*n*(2*n - 1)*(51*n^2 - 162*n + 127)*a(n) = (n-1)*(5457*n^4 - 22791*n^3 + 32144*n^2 - 17536*n + 3072)*a(n-1) + 8*(2*n - 3)*(4*n - 7)*(4*n - 5)*(51*n^2 - 60*n + 16)*a(n-2).
a(n) ~ sqrt(3 + 5/sqrt(17)) * (107 + 51*sqrt(17))^n / (sqrt(Pi*n) * 2^(6*n+2)). (End)
From Peter Bala, Feb 21 2022: (Start)
a(n) = [x^n] ( (1 - x)*(1 - x^2) )^(-n). Cf. A234839.
a(n) = Sum_{k = 0..floor(n/2)} binomial(2*n-2*k-1,n-2*k)*binomial(n+k-1,k).
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 3*x^2 + 9*x^3 + 32*x^4 + 119*x^5 + ... is the g.f. of A063020.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k.
Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and positive integers n and k.
The o.g.f. A(x) is the diagonal of the bivariate rational function 1/(1 - t/((1-x)*(1-x^2))) and hence is an algebraic function over Q(x) by Stanley 1999, Theorem 6.33, p. 197.
Let F(x) = (1/x)*Series_Reversion( x*(1-x)*(1-x^2) ). Then A(x) = 1 + x*d/dx (log(F(x))). (End)
a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(2*n+k-1, k)*binomial(2*n-k-1, n-k). Cf. A352373. - Peter Bala, Jun 05 2024

More terms from Alois P. Heinz, Oct 17 2021

A365751 Expansion of (1/x) * Series_Reversion( x(1+x)(1-x)^3 ).

Original entry on oeis.org

1, 2, 8, 38, 201, 1134, 6688, 40734, 254237, 1617572, 10452416, 68408626, 452530659, 3020870352, 20324167488, 137672551630, 938154745773, 6426806842566, 44234352581896, 305743015718028, 2121318029754770, 14769052147618740, 103148538125870880

Offset: 0

Seiichi Manyama, Sep 18 2023

nonn

Index entries for reversions of series

Cf. A063020, A365752, A365753.

Cf. A352373.

a(n) = sum(k=0, n, (-1)^k*binomial(n+k, k)*binomial(4*n-k+2, n-k))/(n+1);

def A365751(n):
    h = binomial(4*n + 2, n) * hypergeometric([-n, n + 1], [-4 * n - 2], -1) / (n + 1)
    return simplify(h)
print([A365751(n) for n in range(23)])  # Peter Luschny, Sep 20 2023

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(n+k,k) * binomial(4*n-k+2,n-k).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(3*n-2*k+1,n-2*k). - Seiichi Manyama, Jan 18 2024
a(n) = (1/(n+1)) * [x^n] 1/( (1+x) * (1-x)^3 )^(n+1). - Seiichi Manyama, Feb 16 2024

A370100 a(n) = Sum_{k=0..n} binomial(4n,k) binomial(2*n-k-1,n-k).

Original entry on oeis.org

1, 5, 47, 500, 5615, 65005, 767396, 9183144, 110995695, 1351922495, 16566597047, 204010570296, 2522556212228, 31298015910140, 389458822888280, 4858487926378000, 60742838865326319, 760901358321592611, 9547848458062427405, 119990407515367475700

Offset: 0

Seiichi Manyama, Feb 10 2024

Cf. A001448, A352373, A370101, A370102.

Cf. A365754.

Table[Sum[Binomial[4*n, k]*Binomial[2*n - k - 1, n - k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 12 2024 *)

a(n) = sum(k=0, n, binomial(4*n, k)*binomial(2*n-k-1, n-k));

a(n) = [x^n] ( (1+x)^4/(1-x) )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1-x)/(1+x)^4 ). See A365754.
From Peter Bala, Jun 08 2024: (Start)
2*n*(n - 1)*(2*n - 1)*(51*n^2 - 144*n + 100)*a(n) = -(n - 1)*(5457*n^4 - 20865*n^3 + 26366*n^2 - 12172*n + 1560)*a(n-1) + 64*(2*n - 3)*(4*n - 5)*(4*n - 7)*(51*n^2 - 42*n + 7)*a(n-2) with a(0) = 1 and a(1) = 5.
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and positive integers n and r. See A352373 for a more general conjecture. (End)
a(n) ~ sqrt(3 + 5/sqrt(17)) * (51*sqrt(17) - 107)^n / (sqrt(Pi*n) * 2^(3*n + 3/2)). - Vaclav Kotesovec, Jun 12 2024
From Seiichi Manyama, Aug 09 2025: (Start)
a(n) = [x^n] 1/((1-x)^(2*n+1) * (1-2*x)^n).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n,k) * binomial(3*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k-1,k) * binomial(3*n-k,n-k). (End)

A370103 a(n) = Sum_{k=0..n} (-1)^k * binomial(2n+k-1,k) binomial(4*n-k-1,n-k).

Original entry on oeis.org

1, 1, 7, 28, 151, 751, 3976, 20924, 112023, 602182, 3260257, 17724928, 96766072, 529977917, 2910984412, 16027963528, 88440034711, 488918693466, 2707393587802, 15014647096172, 83380131228401, 463593653171495, 2580426581343200, 14377474236172320

Offset: 0

Seiichi Manyama, Feb 10 2024

nonn

Cf. A351857, A370104.
Cf. A352373, A370098, A370101.
Cf. A001700, A348410.
Cf. A365854.

a(n) = sum(k=0, n, (-1)^k*binomial(2*n+k-1, k)*binomial(4*n-k-1, n-k));

a(n, s=2, t=3, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial(u*n, n-s*k));

a(n, s=2, t=2, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((u+1)*n-s*k-1, n-s*k));

a(n) = [x^n] 1/( (1+x)^2 * (1-x)^3 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1+x)^2*(1-x)^3 ). See A365854.
a(n) = Sum_{k=0..floor(n/2)} binomial(3*n+k-1,k) * binomial(n,n-2*k).
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+k-1,k) * binomial(2*n-2*k-1,n-2*k).

A370105 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+k-1,k) * binomial(5*n-k-1,n-k).

Original entry on oeis.org

1, 3, 23, 192, 1687, 15253, 140504, 1311292, 12357015, 117318162, 1120436273, 10752242592, 103596191608, 1001494496863, 9709576926716, 94369011385192, 919175964169623, 8970063281146830, 87685232945278010, 858446087522807784, 8415669293820893937

Offset: 0

Seiichi Manyama, Feb 10 2024

nonn

Cf. A348410, A352373.

Cf. A365752.

a(n) = sum(k=0, n, (-1)^k*binomial(n+k-1, k)*binomial(5*n-k-1, n-k));

a(n, s=2, t=1, u=3) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((u+1)*n-s*k-1, n-s*k));

a(n) = [x^n] 1/( (1+x) * (1-x)^4 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1+x)*(1-x)^4 ). See A365752.
a(n) = Sum_{k=0..floor(n/2)} binomial(n+k-1,k) * binomial(4*n-2*k-1,n-2*k).

cp's OEIS Frontend

A348410 Number of nonnegative integer solutions to n = Sum_{i=1..n} (a_i + b_i), with b_i even.

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A365751 Expansion of (1/x) * Series_Reversion( x*(1+x)*(1-x)^3 ).

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A370100 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(2*n-k-1,n-k).

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A370103 a(n) = Sum_{k=0..n} (-1)^k * binomial(2*n+k-1,k) * binomial(4*n-k-1,n-k).

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A370105 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+k-1,k) * binomial(5*n-k-1,n-k).

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A365751 Expansion of (1/x) * Series_Reversion( x(1+x)(1-x)^3 ).

A370100 a(n) = Sum_{k=0..n} binomial(4n,k) binomial(2*n-k-1,n-k).

A370103 a(n) = Sum_{k=0..n} (-1)^k * binomial(2n+k-1,k) binomial(4*n-k-1,n-k).