A093883 -id:A093883 - cp's OEIS frontend

A203518 a(n) = Product_{2 <= i < j <= n+1} (F(i) + F(j)), where F = A000045 (Fibonacci numbers).

1, 3, 60, 20160, 259459200, 329533940736000, 102591687479575117824000, 20251578856869019790329341542400000, 6518596139761671764183992268499872995344384000000, 8899914870403074273776879003081000194727401271025610417766400000000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

Each term divides its successor, as in A203519. It is conjectured that each term is divisible by the corresponding superfactorial, A000178(n). See A093883 for a guide to related sequences.

Cf. A000045, A203519.

F:= combinat[fibonacci]:
a:= n-> mul(mul(F(i)+F(j), i=2..j-1), j=3..n+1):
seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017

f[j_] := Fibonacci[j + 1]; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}] (* A000178 *)
Table[v[n], {n, 1, z}]                (* A203518 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]   (* A203519 *)
Table[v[n]/d[n], {n, 1, 20}]          (* A203520 *)

a(n) ~ c * d^n * phi^(n^3/3 + n^2/2) / 5^(n^2/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio, d = 3.99077126463315610748163699882855013294148355045548571306491607634698645935... and c = 0.019290318831631524125422284... - Vaclav Kotesovec, Apr 09 2021

Name edited by Alois P. Heinz, Jul 23 2017

A203521 a(n) = Product_{1 <= i < j <= n} (prime(i) + prime(j)).

Original entry on oeis.org

1, 1, 5, 280, 302400, 15850598400, 32867800842240000, 5539460271229108224000000, 55190934927547677562078494720000000, 61965661927377302817151474643396198400000000000, 14512955968670787590604912803260278557019929051136000000000000

Offset: 0

Clark Kimberling, Jan 03 2012

nonn

Each term divides its successor, as in A203511. It is conjectured that each term is divisible by the corresponding superfactorial, A000178(n). See A093883 for a guide to related sequences.

			a(1) = 1.
a(2) = 2 + 3 = 5.
a(3) = (2+3)(2+5)(3+5) = 280.

Alois P. Heinz, Table of n, a(n) for n = 0..32

Cf. A000040, A080358, A203522, A203523, A203524.

a:= n-> mul(mul(ithprime(i)+ithprime(j), i=1..j-1), j=2..n):
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 23 2017

f[j_] := Prime[j]; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}] (* A000178 *)
Table[v[n], {n, 1, z}]                (* A203521 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]   (* A203522 *)
Table[v[n]/d[n], {n, 1, 20}]          (* A203523 *)

Name edited by Alois P. Heinz, Jul 23 2017

A203524 a(n) = Product_{2 <= i < j <= n+1} (prime(i) + prime(j)).

Original entry on oeis.org

1, 8, 960, 3870720, 535088332800, 4746447547269120000, 2251903055463146166681600000, 101133031075657891684280256430080000000, 764075218501479062478490016486870993810227200000000, 510692344365454233151092604262379676645631378616169267200000000000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

Each term divides its successor, as in A203525. It is conjectured that each term is divisible by the corresponding superfactorial, A000178(n); as in A203526. See A093883 for a guide to related sequences.

Cf. A000040, A203315, A203525, A203526.

a:= n-> mul(mul(ithprime(i)+ithprime(j), i=2..j-1), j=3..n+1):
seq(a(n), n=1..10);  # Alois P. Heinz, Jul 23 2017

f[j_] := Prime[j + 1]; z = 17;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}]    (* A000178 *)
Table[v[n], {n, 1, z}]                   (* A203524 *)
Table[v[n + 1]/(8 v[n]), {n, 1, z - 1}]  (* A203525 *)
Table[v[n]/d[n], {n, 1, 20}]             (* A203526 *)

Name edited by Alois P. Heinz, Jul 23 2017

A203748 Vandermonde sequence using x^2 + xy + y^2 applied to (0,1,1,2,2,...,floor(n/2)).

Original entry on oeis.org

1, 1, 3, 588, 1382976, 759365845056, 11257740654368225472, 85256857822344357223236943872, 30991931452969951465382132459004342829056, 10117962915393557751514211466029580457669394910570086400

Offset: 1

Clark Kimberling, Jan 05 2012

nonn

See A093883 for a discussion and guide to related sequences.

f[j_] := Floor[j/2]; z = 15;
u := Product[f[j]^2 + f[j] f[k] + f[k]^2, {j, 1, k - 1}]
v[n_] := Product[u, {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203748 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203749 *)
Table[Sqrt[v[n + 1]/v[n]], {n, 1, z}]
Table[Sqrt[v[2 n]/v[2 n - 1]], {n, 1, z}] (* A203750 *)
Table[Sqrt[v[2 n + 1]/(3 v[2 n])],
   {n, 1, z}]  (* A203751 *)
%/%%         (* A000027 *)

A203755 Vandermonde sequence using x^4 + y^4 applied to (0,1,1,2,2,...,floor(n/2)).

Original entry on oeis.org

1, 1, 2, 9248, 1368408064, 7012482928301113344, 5821608871192502942968054284288, 827078717211493220641742410981240687143117914112, 60161773220249337113595772781004931116549061984924929733289475833856

Offset: 1

Clark Kimberling, Jan 05 2012

nonn

See A093883 for a discussion and guide to related sequences.

f[j_] := Floor[j/2]; z = 16;
u := Product[f[j]^4 + f[k]^4, {j, 1, k - 1}]
v[n_] := Product[u, {k, 2, n}]
Table[v[n], {n, 1, z}]        (* A203755 *)
Table[v[n + 1]/v[n], {n, 1, z}]  (* A203756 *)
Table[Sqrt[v[n + 1]/v[n]], {n, 1, z}]
Table[Sqrt[v[2 n]/v[2 n - 1]], {n, 1, z}]  (* A203757 *)
Table[Sqrt[v[2 n + 1]/(2 v[2 n])],
  {n, 1, z}]  (* A203758 *)
%/%%   (* A000290 *)

A203773 Vandermonde sequence using x^2 + y^2 applied to (0,1,1,2,2,...,floor(n/2)).

Original entry on oeis.org

1, 1, 2, 200, 160000, 24336000000, 66627100800000000, 77020928524800000000000000, 2849158187989401600000000000000000000, 78690953969671659336819671040000000000000000000000

Offset: 1

Clark Kimberling, Jan 05 2012

nonn

See A093883 for a discussion and guide to related sequences.

f[j_] := Floor[j/2]; z = 20;
u := Product[f[j]^2 + f[k]^2, {j, 1, k - 1}]
v[n_] := Product[u, {k, 2, n}]
Table[v[n], {n, 1, z}]         (* A203773 *)
Table[v[n + 1]/v[n], {n, 1, z}]
Table[Sqrt[v[n + 1]/v[n]], {n, 1, z}]
Table[Sqrt[v[2 n]/v[2 n - 1]], {n, 1, z}]  (* A203774 *)
Table[Sqrt[v[2 n + 1]/(2 v[2 n])],
{n, 1, z}]  (* A203775 *)
%/%%          (* A000027 *)

A093884 Product of all possible sums of three numbers taken from among first n natural numbers.

Original entry on oeis.org

6, 3024, 2874009600, 159950125679984640000, 20708778572935434707683938140160000000, 302101709923756073800654275737927385319576932502732800000000000

Offset: 3

Amarnath Murthy, Apr 22 2004

nonn

			a(4) = (1+2+3)*(1+2+4)*(1+3+4)*(2+3+4) = 3024.

Amarnath Murthy, Another combinatorial approach towards generalizing the AM-GM inequality, Octogon Mathematical Magazine Vol. 8, No. 2, October 2000.
Amarnath Murthy, Smarandache Dual Symmetric Functions And Corresponding Numbers Of The Type Of Stirling Numbers Of The First Kind. Smarandache Notions Journal Vol. 11, No. 1-2-3 Spring 2000.

Vaclav Kotesovec, Table of n, a(n) for n = 3..17

Cf. A093883, A306594.

Table[Product[(j + k + m), {k, 2, n}, {j, 1, k - 1}, {m, 1, j - 1}], {n, 3, 10}] (* Vaclav Kotesovec, Aug 31 2023 *)
Table[Product[Sqrt[BarnesG[3*k] * BarnesG[k+2] * Gamma[k/2 + 1] / Gamma[3*k/2]] / (BarnesG[2*k + 1] * 2^((k-1)/2)), {k, 1, n}], {n, 3, 10}] (* Vaclav Kotesovec, Aug 31 2023 *)

a(n) ~ sqrt(Pi/A) * 2^(5/12 - n/4 - n^2 - 2*n^3/3) * 3^(-1/6 - 7*n/24 + 3*n^3/4) * exp(1/24 - n/3 + 3*n^2/4 - 11*n^3/36 + zeta(3)/(48*Pi^2)) * n^(11/24 + n/3 - n^2/2 + n^3/6), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 31 2023

More terms from Vladeta Jovovic, May 27 2004

A203472 a(n) = Product_{3 <= i < j <= n+2} (i + j).

Original entry on oeis.org

1, 7, 504, 498960, 8562153600, 3085457671296000, 27493649380770693120000, 6982164025191299372050022400000, 57286678477842677171688269225656320000000, 16987900892972660430046341200043192304533504000000000, 201504981205067832055356568153709798734509139298353152000000000000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

Each term divides its successor, as in A203470. It is conjectured that each term is divisible by the corresponding superfactorial, A000178(n), as in A203474.

G. C. Greubel, Table of n, a(n) for n = 1..35

Cf. A000178, A093883, A203470, A203473, A203474.

[(&*[(&*[i+j: i in [3..j]])/(2*j): j in [3..n+2]]): n in [1..20]]; // G. C. Greubel, Aug 26 2023

a:= n-> mul(mul(i+j, i=3..j-1), j=4..n+2):
seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017

(* First program *)
f[j_]:= j + 2;    z = 16;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}] (* A000178 *)
Table[v[n], {n,z}]             (* A203472 *)
Table[v[n+1]/v[n], {n,z-1}]    (* A203473 *)
Table[v[n]/d[n], {n,20}]       (* A203474 *)
(* Second program *)
Table[(18*2^(n+2)^2/Pi^(n/2))*BarnesG[n+3]*BarnesG[n+7/2]/(BarnesG[n+ 6]*BarnesG[7/2]), {n,20}] (* G. C. Greubel, Aug 26 2023 *)

[product( gamma(2*j)/gamma(j+3) for j in range(3,n+3) ) for n in range(1,20)] # G. C. Greubel, Aug 26 2023

a(n) ~ 3*sqrt(A) * 2^(n^2 + 9*n/2 + 185/24) * n^(n^2/2 - n/2 - 179/24) / (Pi^(3/2) * exp(3*n^2/4 - n/2 + 1/24)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 09 2021
From G. C. Greubel, Aug 26 2023: (Start)
a(n) = Prod_{j=3..n+2} Prod_{i=3..j-1} (i + j).
a(n) = Prod_{j=3..n+2} Gamma(2*j)/Gamma(j+3).
a(n) = (18*2^(n+2)^2/Pi^(n/2))*BarnesG(n+3)*BarnesG(n+7/2)/(BarnesG(n+ 6)*BarnesG(7/2)). (End)

Name edited by Alois P. Heinz, Jul 23 2017

A203473 a(n) = v(n+1)/v(n), where v=A203472.

Original entry on oeis.org

7, 72, 990, 17160, 360360, 8910720, 253955520, 8204716800, 296541907200, 11861676288000, 520431047136000, 24858235898496000, 1284342188088960000, 71382386874839040000, 4247252019052922880000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..250

Cf. A093883, A203472, A203474.

[Floor(Gamma(2*n+6)/Gamma(n+6)): n in [1..16]]; // G. C. Greubel, Aug 27 2023

(* First program *)
f[j_]:= j+2; z=16;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}];
d[n_]:= Product[(i-1)!, {i,n}]  (* A000178 *)
Table[v[n], {n,z}]              (* A203472 *)
Table[v[n+1]/v[n], {n,z-1}]     (* this sequence *)
Table[v[n]/d[n], {n,20}]        (* A203474 *)
(* Second program *)
Table[Pochhammer[n+6,n], {n,20}] (* G. C. Greubel, Aug 27 2023 *)

[rising_factorial(n+6, n) for n in range(1,16)] # G. C. Greubel, Aug 27 2023

a(n) ~ 2^(2*n + 11/2) * n^n / exp(n). - Vaclav Kotesovec, Apr 09 2021
a(n) = RisingFactorial(6 + n, n). - Peter Luschny, Mar 22 2022
Since v(n) = (135/4)*(2^(n+2)^2/Pi^(n/2))*(BarnesG(n+3)*BarnesG(n+7/2) )/( BarnesG(9/2)*BarnesG(n+6) ) then v(n+1)/v(n) = Gamma(2*n+6) / Gamma(n+6). - G. C. Greubel, Aug 27 2023

A203479 a(n) = Product_{1 <= i < j <= n} (2^i + 2^j - 2).

Original entry on oeis.org

1, 4, 320, 2027520, 3855986196480, 8359491805553413324800, 79457890145647634305213865656320000, 12897878211365028383150895090566532213003150950400000, 140613650417826346093374124598539442743630963394643403845144815232614400000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

Each term divides its successor, as in A203480.

G. C. Greubel, Table of n, a(n) for n = 1..21

Cf. A000225, A093883, A203305, A203480, A203481.

[(&*[(&*[2^j+2^k-2: k in [1..j]])/(2^(j+1)-2): j in [1..n]]): n in [1..15]]; // G. C. Greubel, Aug 28 2023

a:= n-> mul(mul(2^i+2^j-2, i=1..j-1), j=2..n):
seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017

(* First program *)
f[j_]:= 2^j -1; z = 15;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]               (* A203479 *)
Table[v[n+1]/v[n], {n,z-1}]      (* A203480 *)
Table[v[n+1]/(4*v[n]), {n,z-1}]  (* A203481 *)
(* Second program *)
Table[Product[2^j +2^k -2, {j,n}, {k,j-1}], {n,15}] (* G. C. Greubel, Aug 28 2023 *)

[product(product(2^j+2^k-2 for k in range(1,j)) for j in range(1,n+1)) for n in range(1,16)] # G. C. Greubel, Aug 28 2023

a(n) ~ c * 2^((n-1)*n*(n+1)/3) / QPochhammer(1/2, 1/4)^(n-1), where c = 0.0732262905669624786298393270254722268761908164083517721484477901776137... - Vaclav Kotesovec, Aug 09 2025

Name edited by Alois P. Heinz, Jul 23 2017

cp's OEIS Frontend

A203518 a(n) = Product_{2 <= i < j <= n+1} (F(i) + F(j)), where F = A000045 (Fibonacci numbers).

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A203521 a(n) = Product_{1 <= i < j <= n} (prime(i) + prime(j)).

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A203524 a(n) = Product_{2 <= i < j <= n+1} (prime(i) + prime(j)).

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A203748 Vandermonde sequence using x^2 + xy + y^2 applied to (0,1,1,2,2,...,floor(n/2)).

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A203755 Vandermonde sequence using x^4 + y^4 applied to (0,1,1,2,2,...,floor(n/2)).

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A203773 Vandermonde sequence using x^2 + y^2 applied to (0,1,1,2,2,...,floor(n/2)).

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A093884 Product of all possible sums of three numbers taken from among first n natural numbers.

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A203472 a(n) = Product_{3 <= i < j <= n+2} (i + j).

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