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A333592
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a(n) = Sum_{k = 0..n} binomial(n + k - 1, k)^2.
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5
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1, 2, 14, 146, 1742, 22252, 296438, 4063866, 56884430, 808970960, 11649069764, 169444272692, 2485268015414, 36707034407396, 545386280953262, 8144809577111146, 122177689609022670, 1839933272106181720, 27804610617723365072, 421476329309967621504, 6406685024966332359492
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OFFSET
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0,2
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COMMENTS
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Compare with the closed-form evaluation Sum_{k = 0..n} C(n+k-1,k) = C(2*n,n) = A000984(n) (see the first comment in A001700 for a proof).
It is well-known that Sum_{k = 0..n} C(n,k)^2 = C(2*n,n). Here, we consider by analogy Sum_{k = 0..n} C(-n,k)^2, where C(-n,k) = (-1)^k*C(n+k-1,k) for integer n and nonnegative integer k.
The sequence b(n) = C(2*n,n) of central binomial coefficients satisfies the congruences b(n*p^k) = b(n*p^(k-1)) ( mod p^(3*k) ) for all prime p >= 5 and any positive integers n and k - see Mestrovic. We conjecture that the present sequence also satisfies these congruences. Some examples of the congruences are given below.
More generally, calculation suggests that for positive integer A and integer B, the sequence a(A,B;n) := Sum_{k = 0..A*n} C(B*n+k-1,k)^2 may satisfy the same congruences.
The sequence (a(p) - 2)/(2*p^3) for prime p >= 5 begins [89, 5924, 63652995, 8353899501, 187251503369243, 30724327840061789, 937835335872800013431, ...]. Cf. A034602.
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LINKS
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FORMULA
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EXAMPLE
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Examples of congruences:
a(11) - a(1) = 169444272692 - 2 = 2*(3^2)*5*7*(11^3)*397*509 == 0 ( mod 11^3 ).
a(2*7) - a(2) = 545386280953262 - 14 = (2^5)*(3^2)*(7^4)*788714021 == 0 ( mod 7^3 ).
a(5^2) - a(5) = 5375188503768783714940459752 - 22252 = (2^2)*(5^6)*(31^2)* 89493252924350197127 == 0 ( mod 5^6 ).
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MAPLE
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seq( add( binomial(n+k-1, k)^2, k = 0..n ), n = 0..25);
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MATHEMATICA
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Table[Binomial[2*n-1, n]^2 * HypergeometricPFQ[{1, -n, -n}, {1 - 2*n, 1 - 2*n}, 1], {n, 1, 20}] (* Vaclav Kotesovec, Mar 28 2020 *)
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PROG
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(PARI) a(n) = sum(k=0, n, binomial(n+k-1, k)^2); \\ Michel Marcus, Mar 29 2020.
(Python)
from math import comb
def A333592(n): return sum(comb(n+k-1, k)**2 for k in range(n+1)) if n else 1 # Chai Wah Wu, Oct 28 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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