A361608 -id:A361608 - cp's OEIS frontend

A361609 a(n) = 4^n(1 + (23/8)n + (9/8)*n^2).

1, 20, 180, 1264, 7808, 44544, 240640, 1249280, 6291456, 30932992, 149159936, 707788800, 3313500160, 15334375424, 70262980608, 319169757184, 1438814044160, 6442450944000, 28673201668096, 126924873531392, 559101662724096, 2451910929940480, 10709243254538240, 46601700831657984

Offset: 0

R. J. Mathar, Mar 17 2023

The sequences A(n,k) = Sum_{j=0..n} Sum_{i=0..j} (-1)^(j-i) * binomial(n,j) *binomial(j,i) * binomial(j+k+(k+1)*i,j+k) are C-sequences for fixed integer k, here A(n,k=2) = a(n).

Winston de Greef, Table of n, a(n) for n = 0..1640
Project Euler, Problem 831. Triple Product
Index entries for linear recurrences with constant coefficients, signature (12,-48,64).

Cf. A027471 (k=1), A361610 (k=3), A361608 (k=5).

LinearRecurrence[{12, -48, 64}, {1, 20, 180}, 25] (* or *)
A361609[n_] := 4^n (1 + 23/8 n + 9/8 n^2);
Array[A361609, 25, 0] (* Paolo Xausa, Jan 18 2024 *)

def A361609(n): return (n*(9*n + 23) + 8)<<((n<<1)-3) if n > 1 else 19*n+1 # Chai Wah Wu, Mar 17 2023

G.f.: ( -1-8*x+12*x^2 ) / (4*x-1)^3.
a(n) = 12*a(n-1) -48*a(n-2) +64*a(n-3).
D-finite with recurrence (-9*n^2-5*n+6)*a(n) +4*(9*n^2+23*n+8)*a(n-1)=0.

A361610 a(n) = 5^n(n+1)(4n^2+14n+3)/3.

Original entry on oeis.org

1, 70, 1175, 13500, 128125, 1081250, 8421875, 61875000, 434765625, 2949218750, 19443359375, 125195312500, 790283203125, 4904785156250, 29998779296875, 181152343750000, 1081695556640625, 6394958496093750, 37471771240234375, 217819213867187500, 1257038116455078125

Offset: 0

R. J. Mathar, Mar 17 2023

The sequences A(n,k) = Sum_{j=0..n} Sum_{i=0..j} (-1)^(j-i) * binomial(n,j) * binomial(j,i) * binomial(j+k+(k+1)*i,j+k) are C-sequences for fixed integer k, here A(n,k=3) = a(n).

Winston de Greef, Table of n, a(n) for n = 0..1408
Project Euler, Problem 831. Triple Product
Index entries for linear recurrences with constant coefficients, signature (20,-150,500,-625).

Cf. A027471 (k=1), A361609 (k=2), A361608 (k=5).

LinearRecurrence[{20,-150,500,-625},{1,70,1175,13500},30] (* Harvey P. Dale, Aug 29 2024 *)

def A361610(n): return 5**n*(n*(n*(4*n + 18) + 17) + 3)//3 # Chai Wah Wu, Mar 17 2023

G.f.: (1 + 50*x - 75*x^2) / (5*x - 1)^4.
a(n) = 20*a(n-1) -150*a(n-2) +500*a(n-3) -625*a(n-4).
D-finite with recurrence n*(4*n^2+6*n-7)*a(n) -5*(n+1)*(4*n^2+14*n+3)*a(n-1)=0.

cp's OEIS Frontend

A361609 a(n) = 4^n(1 + (23/8)n + (9/8)*n^2).

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A361610 a(n) = 5^n(n+1)(4n^2+14n+3)/3.

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cp's OEIS Frontend

A361609 a(n) = 4^n*(1 + (23/8)*n + (9/8)*n^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

A361610 a(n) = 5^n*(n+1)*(4*n^2+14*n+3)/3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

A361609 a(n) = 4^n(1 + (23/8)n + (9/8)*n^2).

A361610 a(n) = 5^n(n+1)(4n^2+14n+3)/3.