A119363 -id:A119363 - cp's OEIS frontend

A119335 Number triangle T(n,k) = Sum_{j=0..n-k} C(k,3j)*C(n-k,3j).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 11, 17, 11, 1, 1, 1, 1, 1, 1, 21, 41, 41, 21, 1, 1, 1, 1, 1, 1, 36, 81, 101, 81, 36, 1, 1, 1, 1, 1, 1, 57, 141, 201, 201, 141, 57, 1, 1, 1

Offset: 0

Paul Barry, May 14 2006

Row sums are A119336. Product of Pascal's triangle and A119337.

			Triangle begins
1;
1, 1;
1, 1, 1;
1, 1, 1,  1;
1, 1, 1,  1,  1;
1, 1, 1,  1,  1,   1;
1, 1, 1,  2,  1,   1,  1;
1, 1, 1,  5,  5,   1,  1,  1;
1, 1, 1, 11, 17,  11,  1,  1, 1;
1, 1, 1, 21, 41,  41, 21,  1, 1, 1;
1, 1, 1, 36, 81, 101, 81, 36, 1, 1, 1;

Seiichi Manyama, Rows n = 0..139, flattened

T(2n,n) gives A119363.

Cf. A119326.

T[n_, k_] := Sum[Binomial[k, 3j] Binomial[n-k, 3j], {j, 0, n-k}];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 14 2023 *)

Column k has g.f. (x^k/(1-x)) * Sum_{j=0..k} C(k,3j)(x/(1-x))^(3j).

More terms from Seiichi Manyama, Mar 12 2019

A139469 a(n) = Sum_{k=0..n} C(n,3k+2)^2.

Original entry on oeis.org

0, 0, 1, 9, 36, 101, 261, 882, 3921, 17253, 67554, 243695, 876789, 3324906, 13166791, 52301709, 203824548, 782913717, 3010327497, 11695756698, 45823049817, 179787741723, 703527078258, 2747647985241, 10739885115573, 42082084255050, 165225573240651

Offset: 0

N. J. A. Sloane, Jun 12 2008

nonn

The recurrence is same as for A119363. - Vaclav Kotesovec, Mar 12 2019

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A024495, A119363, A139468.

[&+[Binomial(n, 3*k+2)^2: k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, Mar 14 2019

Table[Sum[Binomial[n, 3*k + 2]^2, {k, 0, n}], {n, 0, 40}] (* Vaclav Kotesovec, Mar 12 2019 *)

a(n) = sum(k=0, n, binomial(n, 3*k+2)^2); \\ Michel Marcus, Mar 12 2019

a(n) ~ 4^n / (3*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 12 2019

A139468 a(n) = Sum{k=0..n} C(n,3k+1)^2.

Original entry on oeis.org

0, 1, 4, 9, 17, 50, 261, 1275, 5028, 17253, 58601, 218042, 876789, 3537847, 13783018, 52301709, 198627921, 767778786, 3010327497, 11824753551, 46200429186, 179787741723, 700285942731, 2738134757118, 10739885115573, 42164261091351, 165467386466802

Offset: 0

N. J. A. Sloane, Jun 12 2008

nonn

The recurrence is same as for A119363. - Vaclav Kotesovec, Mar 12 2019

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A024494, A119363, A139469.

Table[Sum[Binomial[n,3k+1]^2,{k,0,n}],{n,0,30}] (* Harvey P. Dale, Sep 08 2018 *)

a(n) ~ 4^n / (3*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 12 2019

A307091 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n,2*k)^2.

Original entry on oeis.org

1, 1, 0, -8, -34, -74, 0, 736, 3334, 7606, 0, -80464, -372436, -864772, 0, 9400192, 43976774, 103061158, 0, -1137528688, -5355697084, -12623082284, 0, 140697113792, 665238165916, 1574005263676, 0, -17663830073504, -83769667651816, -198760191043784, 0

Offset: 0

Seiichi Manyama, Mar 24 2019

Seiichi Manyama, Table of n, a(n) for n = 0..1879

Central coefficients of number triangle A307090.

Cf. A000984, A119358, A119363.

Table[Sum[(-1)^k*Binomial[n, 2*k]^2, {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vaclav Kotesovec, Mar 24 2019 *)
Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1/2, 1/2, 1}, -1], {n, 0, 30}] (* Vaclav Kotesovec, Mar 24 2019 *)

{a(n) = sum(k=0, n\2, (-1)^k*binomial(n, 2*k)^2)}

a(4*n+2) = 0 for n >= 0.
From Peter Bala, Mar 17 2023: (Start)
n*(n-1)*(6*n^2-24*n+23)a(n) = 4*(n-1)*(2*n-3)*(3*n^2-9*n+4)*a(n-1) - 4*(3*n^2-9*n+4)*(2*n-3)^2*a(n-2) - 8*(n-2)*(2*n-3)*(3*n^2-9*n+4)*a(n-3) - 4*(n-2)*(n-3)*(6*n^2-12*n+5)*a(n-4) with a(0) = 1, a(1) = 1, a(2) = 0 and a(3) = -8.
a(n) = hypergeom([(1-n)/2, (1-n)/2, -n/2, -n/2], [1/2, 1/2, 1], -1).
Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for positive integers n and r and all primes p >= 3. (End)

A307158 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n,3*k)^2.

Original entry on oeis.org

1, 1, 1, 0, -15, -99, -398, -1175, -2351, 0, 29601, 183195, 756978, 2351805, 4885791, 0, -63746991, -400000275, -1675991918, -5274560891, -11081420615, 0, 147257373891, 931226954949, 3929550225586, 12446852889901, 26304183607651, 0, -353181028924809

Offset: 0

Seiichi Manyama, Mar 27 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1752

Central coefficients of number triangle A307156.

Cf. A057681, A119363, A307091.

a[n_] := Sum[(-1)^k * Binomial[n,3*k]^2, {k, 0, Floor[n/3]}]; Array[a, 30, 0] (* Amiram Eldar, May 20 2021 *)

{a(n) = sum(k=0, n\3, (-1)^k*binomial(n, 3*k)^2)}

a(6*n+3) = 0 for n >= 0.

A119364 Central coefficients of number triangle A119335.

Original entry on oeis.org

0, 1, 1, 1, 11, 81, 351, 1149, 3529, 12601, 52724, 222641, 879308, 3295384, 12303201, 47320365, 186738507, 739129809, 2894481813, 11237844615, 43647142533, 170543919327, 669744238998, 2633027605209, 10337488816041, 40544676533466

Offset: 0

Paul Barry, May 16 2006

A119363(n)-a(n)=A119365(n).

a(n)=sum{k=0..n-1, C(n+1,3k)*C(n-1,3k)}; a(n)=A119335(2n,n+1).

cp's OEIS Frontend

A119335 Number triangle T(n,k) = Sum_{j=0..n-k} C(k,3j)*C(n-k,3j).

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A139469 a(n) = Sum_{k=0..n} C(n,3k+2)^2.

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A139468 a(n) = Sum{k=0..n} C(n,3k+1)^2.

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A307091 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n,2*k)^2.

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A307158 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n,3*k)^2.

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A119364 Central coefficients of number triangle A119335.

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