A361609 -id:A361609 - cp's OEIS frontend

A361608 a(n) = 7^n(n+1)(81n^4+684n^3+1401n^2+434n+40)/40.

1, 924, 48804, 1337014, 26622288, 437049228, 6295986235, 82489361052, 1005444707211, 11576481361732, 127278262644918, 1346951022678114, 13803666582387682, 137633164619393268, 1340161331495822661, 12782144706910135480, 119711031072135899781, 1103157160378734314700, 10019811250265958667288

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=5) = a(n).

Project Euler, Problem 831. Triple Product
R. J. Mathar, On an alternating double sum of a triple product of aerated binomial coefficients, arXiv:2306.08022 (2023)
Index entries for linear recurrences with constant coefficients, signature (42,-735,6860,-36015,100842,-117649).

Cf. A027471 (k=1), A361609 (k=2), A361610 (k=3).

LinearRecurrence[{42,-735,6860,-36015,100842,-117649},{1,924,48804,1337014,26622288,437049228},20] (* Harvey P. Dale, May 29 2023 *)

def A361608(n): return 7**n*(n*(n*(n*(n*(81*n + 765) + 2085) + 1835) + 474) + 40)//40 # Chai Wah Wu, Mar 17 2023

G.f.: ( 1+882*x+10731*x^2-40474*x^3+36015*x^4 ) / (7*x-1)^6 .
a(n) = +42*a(n-1) -735*a(n-2) +6860*a(n-3) -36015*a(n-4) +100842*a(n-5) -117649*a(n-6).
D-finite with recurrence n*(81*n^4+360*n^3-165*n^2-640*n+404)*a(n) -7*(n+1)*(81*n^4+684*n^3+1401*n^2+434*n+40)*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

A361608 a(n) = 7^n(n+1)(81n^4+684n^3+1401n^2+434n+40)/40.

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

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cp's OEIS Frontend

A361608 a(n) = 7^n*(n+1)*(81*n^4+684*n^3+1401*n^2+434*n+40)/40.

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

A361608 a(n) = 7^n(n+1)(81n^4+684n^3+1401n^2+434n+40)/40.

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