A101686 -id:A101686 - cp's OEIS frontend

A268434 Triangle read by rows, Lah numbers of order 2, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+((n-1)^2+k^2)*T(n-1,k), for n>=0 and 0<=k<=n.

1, 0, 1, 0, 2, 1, 0, 10, 10, 1, 0, 100, 140, 28, 1, 0, 1700, 2900, 840, 60, 1, 0, 44200, 85800, 31460, 3300, 110, 1, 0, 1635400, 3476200, 1501500, 203060, 10010, 182, 1, 0, 81770000, 185874000, 90563200, 14700400, 943800, 25480, 280, 1

Offset: 0

Peter Luschny, Mar 07 2016

0

			[1]
[0,        1]
[0,        2,         1]
[0,       10,        10,        1]
[0,      100,       140,       28,        1]
[0,     1700,      2900,      840,       60,      1]
[0,    44200,     85800,    31460,     3300,    110,     1]
[0,  1635400,   3476200,  1501500,   203060,  10010,   182,   1]

Peter Luschny, The P-transform.

Cf. A038207 (order 0), A111596 (order 1), A269946 (order 3).

Cf. A036969, A105278, A204579, A269938, A269941, A269944, A269945.

T := proc(n,k) option remember;
if n=k then return 1 fi; if k<0 or k>n then return 0 fi;
T(n-1,k-1)+((n-1)^2+k^2)*T(n-1,k) end:
seq(seq(T(n,k), k=0..n), n=0..8);
# Alternatively with the P-transform (cf. A269941):
A268434_row := n -> PTrans(n, n->`if`(n=1,1, ((n-1)^2+1)/(n*(4*n-2))),
(n,k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A268434_row(n)), n=0..8);

T[n_, n_] = 1; T[, 0] = 0; T[n, k_] /; 0 < k < n := T[n, k] = T[n-1, k-1] + ((n-1)^2 + k^2)*T[n-1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2017 *)

#cached_function
def T(n, k):
    if n==k: return 1
    if k<0 or k>n: return 0
    return T(n-1, k-1)+((n-1)^2+k^2)*T(n-1, k)
for n in range(8): print([T(n, k) for k in (0..n)])
# Alternatively with the function PtransMatrix (cf. A269941):
PtransMatrix(8, lambda n: 1 if n==1 else ((n-1)^2+1)/(n*(4*n-2)), lambda n, k: (-1)^k*factorial(2*n)/factorial(2*k))

T(n,k) = (-1)^k*((2*n)!/(2*k)!)*P[n,k](s(n)) where P is the P-transform and s(n) = ((n-1)^2+1)/(n*(4*n-2)). The P-transform is defined in the link. Compare also the Sage and Maple implementations below.
T(n,k) = Sum_{j=k..n} A269944(n,j)*A269945(j,k).
T(n,1) = Product_{k=1..n} (k-1)^2+1 for n>=1 (cf. A101686).
T(n,n-1) = (n-1)*n*(2*n-1)/3 for n>=1 (cf. A006331).
Row sums: A269938.

A277352 a(n) = Product_{k=1..n} (2*k^2+1).

Original entry on oeis.org

1, 3, 27, 513, 16929, 863379, 63026667, 6239640033, 804913564257, 131200910973891, 26371383105752091, 6408246094697758113, 1851983121367652094657, 627822278143634060088723, 246734155310448185614868139, 111277104045012131712305530689

Offset: 0

Vaclav Kotesovec, Oct 10 2016

nonn

Guadalupe proves that a(n) is not square for n > 0. - Charles R Greathouse IV, Mar 16 2023

Russelle Guadalupe, Squares of the form Product_{k=1..n} (2k^2+l) with l odd, arXiv:2201.00501 [math.NT], 2022.

Cf. A101686, A277347, A277353, A277354.

Table[Product[2*k^2+1, {k, 1, n}], {n, 0, 15}]

a(n)=prod(k=1,n,2*k^2+1) \\ Charles R Greathouse IV, Mar 16 2023

a(n) ~ 2^(n+3/2) * n^(2*n+1) * sinh(Pi/sqrt(2)) / exp(2*n).

A277353 a(n) = Product_{k=1..n} (3*k^2+1).

Original entry on oeis.org

1, 4, 52, 1456, 71344, 5422144, 591013696, 87470027008, 16881715212544, 4119138511860736, 1239860692070081536, 451309291913509679104, 195416923398549691052032, 99271797086463243054432256, 58471088483926850159060598784, 39526455815134550707524964777984

Offset: 0

Vaclav Kotesovec, Oct 10 2016

nonn

Cf. A101686, A277352, A277354.

Table[Product[3*k^2+1, {k, 1, n}], {n, 0, 15}]

a(n) ~ 2*3^(n+1/2) * n^(2*n+1) * sinh(Pi/sqrt(3)) / exp(2*n).

A354054 a(n) = Product_{k=0..n} (k^6 + 1).

Original entry on oeis.org

1, 2, 130, 94900, 388805300, 6075471617800, 283463279271694600, 33349454806314869690000, 8742392830201411514885050000, 4646074730467898538293540742100000, 4646079376542629006192079035640742100000, 8230817672466612927467651920537784356160200000

Offset: 0

Vaclav Kotesovec, May 16 2022

nonn

Winston de Greef, Table of n, a(n) for n = 0..103

Cf. A002604, A101686, A255433, A255434, A258871, A354052.

A354054 := proc(n)
    mul( k^6+1,k=0..n) ;
end proc:
seq(A354054(n),n=0..40) ; # R. J. Mathar, Jul 17 2023

Table[Product[k^6 + 1, {k, 0, n}], {n, 0, 15}]

PARI
```
a(n) = prod(k=1, n, k^6+1);
```

a(n) ~ (2*sinh(2*Pi) - 4*sinh(Pi)*cos(sqrt(3)*Pi)) * n^(6*n + 3) / exp(6*n).

a(n) ~ A258871 * (n!)^6.

A228120 a(n) = (1^2 + 1)(2^2 + 1)(3^2 + 1)...(((prime(n) - 1)/2)^2 + 1).

Original entry on oeis.org

2, 10, 100, 44200, 1635400, 5315050000, 435834100000, 5370347780200000, 26078677338040210000000, 5893781078397087460000000, 142760638726203851727985000000000, 20723419838773203524537758570000000000, 9159751568737755957845689287940000000000, 2354514140744040168964234431464977000000000000

Offset: 2

Zhi-Wei Sun, Aug 11 2013

nonn

The author has shown that a(n) == 2 (mod p_n) if p_n == 3 (mod 4). He has also established the following general theorem:
Let p be any odd prime and let d be any quadratic non-residue modulo p. Then we have the congruence
Product_{x=1..(p-1)/2} (x^2 - d) == 2*(-1)^((p+1)/2) (mod p).
This can be proved as follows: By Wilson's theorem we have (((p-1)/2)!)^2 == (-1)^((p+1)/2) (mod p), and thus we reduce the desired congruence to
       Product_{0  Clearly
        Product_{1     == Product_{j=2..(p-1)/2} (1 - j^2)
      = (-1)^((p+1)/2)*(((p-1)/2)!)^2*(p+1)/(2*p-2)
     == -1/2 (mod p),
and Product_{k=2..p-1} (1 - k) = (-1)^(p-2)*(p-2)! == -1 (mod p) by Wilson's theorem. Therefore (*) follows.

			a(3) = (1^1+1)*(2^2+1)*(3^2+1) = 100 and a(3) == 2 (mod 7).

Zhi-Wei Sun, Table of n, a(n) for n = 2..30

Cf. A101686, A000040.

a[n_]:=Product[(x^2+1),{x,1,(Prime[n]-1)/2}]
Table[a[n],{n,2,15}]

A262001 G.f.: 1/(1 - xF'(x)/F(x)) where F(x) = Sum_{n>=0} x^n/n!Product_{k=1..n} (k^2 + 1).

Original entry on oeis.org

1, 2, 10, 60, 400, 2900, 22700, 191600, 1746400, 17230000, 184348000, 2140118000, 26925784000, 366118706000, 5359236310000, 84077608400000, 1407341155720000, 25027454132360000, 471046698018440000, 9351091483806800000, 195213433887227200000, 4274234604872786800000, 97924306054031604400000

Offset: 0

Paul D. Hanna, Sep 08 2015

nonn

Cf. A262002, which is defined by: Sum_{n>=0} x^n/n! * Product_{k=1..n} (k^2+1) = exp( Sum_{n>=1} A262002(n)*x^n/n ).

Sum of all terms results in the 10-adic number x = ...5211383820350605156083728207423149062180073.

			O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 60*x^3 + 400*x^4 + 2900*x^5 + 22700*x^6 +...
where
1 - 1/A(x) =  2*x + 6*x^2 + 28*x^3 + 164*x^4 + 1132*x^5 + 8916*x^6 + 78608*x^7 + 765904*x^8 + 8170752*x^9 +...+ A262002(n)*x^n +...
Note that if we define the logarithmic series:
L(x) = 2*x + 6*x^2/2 + 28*x^3/3 + 164*x^4/4 + 1132*x^5/5 + 8916*x^6/6 + 78608*x^7/7 + 765904*x^8/8 +...+ A262002(n)*x^n/n +...
then exp(L(x)) = 1 + 2*x + 10*x^2/2! + 100*x^3/3! + 1700*x^4/4! + 44200*x^5/5! + 1635400*x^6/6! +...+ A101686(n)*x^n/n! +... where A101686(n) = Product_{k=1..n} (k^2+1).

Cf. A262002.

{a(n) = local(A=1,L=log(sum(m=0,n+1,x^m/m!*prod(k=1,m,k^2+1)) +x*O(x^n))); A=1/(1 - x*L'); polcoeff(A +x*O(x^n), n)}
for(n=0,30,print1(a(n),", "))

G.f.: 1/(1 - G(x)) where G(x) is an o.g.f. of A262002.

a(n) == 0 (mod 10) for n>1.

A262002 L.g.f.: log( Sum_{n>=0} x^n/n! * Product_{k=1..n} (k^2 + 1) ).

Original entry on oeis.org

2, 6, 28, 164, 1132, 8916, 78608, 765904, 8170752, 94755776, 1187551088, 16004096144, 230910861872, 3553052000336, 58100442762368, 1006457051734784, 18415695160624192, 354980774078690496, 7190981550797464448, 152744987909458781824, 3395058926880381635712, 78814259879097446800256

Offset: 1

Paul D. Hanna, Sep 08 2015

nonn

			L.g.f.: L(x) = 2*x + 6*x^2/2 + 28*x^3/3 + 164*x^4/4 + 1132*x^5/5 + 8916*x^6/6 + 78608*x^7/7 + 765904*x^8/8 + 8170752*x^9/9 + 94755776*x^10/10 +...
such that
exp(L(x)) = 1 + 2*x + 10*x^2/2! + 100*x^3/3! + 1700*x^4/4! + 44200*x^5/5! + 1635400*x^6/6! +...+ A101686(n)*x^n/n! +...
where A101686(n) = Product_{k=1..n} (k^2+1).
Also, given the o.g.f. A(x) = Sum_{n>=1} a(n)*x^n,
o.g.f.: A(x) = 2*x + 6*x^2 + 28*x^3 + 164*x^4 + 1132*x^5 + 8916*x^6 +...
then
1/(1 - A(x)) = 1 + 2*x + 10*x^2 + 60*x^3 + 400*x^4 + 2900*x^5 + 22700*x^6 + 191600*x^7 + 1746400*x^8 + 17230000*x^9 + 184348000*x^10 +...+ A262001(n)*x^n +...

Cf. A101686, A262001.

{a(n) = n*polcoeff( log(sum(m=0,n+1,x^m/m!*prod(k=1,m,k^2+1)) +x*O(x^n)), n)}
for(n=1,30,print1(a(n),", "))

O.g.f.: 1 - 1/G(x) where G(x) is the g.f. of A262001.

A325050 a(n) = Product_{k=0..n} (k!^2 + 1).

Original entry on oeis.org

2, 4, 20, 740, 426980, 6148938980, 3187616116170980, 80970552724144881738980, 131634021973939424914920841290980, 17333817381151204925617274632152908873802980, 228254990993381085562170532497621436371926846785405002980

Offset: 0

Vaclav Kotesovec, Mar 26 2019

nonn

Cf. A055209, A101686, A217757, A238695, A325048.

Table[Product[k!^2 + 1, {k, 0, n}], {n, 0, 12}]
Table[BarnesG[n+2]^2 * Product[1 + 1/k!^2, {k, 0, n}], {n, 0, 12}]

a(n) ~ c * n^(n^2 + 2*n + 5/6) * (2*Pi)^(n+1) / (A^2 * exp(3*n^2/2 + 2*n - 1/6)), where c = Product_{k>=0} (1 + 1/k!^2) = 5.1481781945902396880952694880498895... and A is the Glaisher-Kinkelin constant A074962.

A271266 a(n) = Product_{k=1..n} (k^2 + 21).

Original entry on oeis.org

1, 22, 550, 16500, 610500, 28083000, 1600731000, 112051170000, 9524349450000, 971483643900000, 117549520911900000, 16692031969489800000, 2754185274965817000000, 523295202243505230000000, 113555058886840634910000000, 27934544486162796187860000000, 7737868822667094544037220000000

Offset: 0

Michel Marcus, Apr 03 2016

nonn

Yin et al. prove that a(n) is never a square for n > 0.

Q. Yin, Q. Tan and Y. Luo, p-Adic Valuation of (12+21)...(n2+21) and Applications, Southeast Asian Bulletin of Mathematics, Vol. 39(5) (2015) page: 747-754.

Cf. A101686, A241851.

Table[Product[k^2 + 21, {k, n}], {n, 0, 16}] (* Michael De Vlieger, Apr 03 2016 *)

PARI
```
a(n) = prod(k=1, n, (k^2+21));
```

A324442 a(n) = Product_{i=1..n, j=1..n} (i^2 + j).

Original entry on oeis.org

1, 2, 180, 6652800, 402265543680000, 109211487076824381849600000, 295382703175843424854047228769075200000000, 15385012566245626089929288743828190926813939944652800000000000

Offset: 0

Vaclav Kotesovec, Feb 28 2019

nonn

Cf. A079478, A101686, A255322, A324403.

a:= n-> mul(mul(i^2+j, i=1..n), j=1..n):
seq(a(n), n=0..8);  # Alois P. Heinz, Jun 24 2023

Table[Product[i^2 + j, {i, 1, n}, {j, 1, n}], {n, 1, 10}]
Table[Product[Pochhammer[1 + i^2, n], {i, 1, n}], {n, 1, 10}]

From Vaclav Kotesovec, Dec 27 2023: (Start)
a(n) ~ c * n^(2*n^2 + n/2 - 1/4) / exp(2*n^2 - 2*Pi*n^(3/2)/3 - Pi*sqrt(n)/2), where c = 0.31906...
For n>1, a(n) = a(n-1) * Gamma(n - i*sqrt(n)) * Gamma(n + i*sqrt(n)) * Gamma(n^2 + n + 1) * sinh(Pi*sqrt(n)) / (Pi * n^(5/2) * Gamma(n^2)), where i is the imaginary unit. (End)

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

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cp's OEIS Frontend

A268434 Triangle read by rows, Lah numbers of order 2, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+((n-1)^2+k^2)*T(n-1,k), for n>=0 and 0<=k<=n.

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A277352 a(n) = Product_{k=1..n} (2*k^2+1).

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A277353 a(n) = Product_{k=1..n} (3*k^2+1).

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Mathematica

Formula

A354054 a(n) = Product_{k=0..n} (k^6 + 1).

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Mathematica

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A228120 a(n) = (1^2 + 1)*(2^2 + 1)*(3^2 + 1)*...*(((prime(n) - 1)/2)^2 + 1).

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A262001 G.f.: 1/(1 - x*F'(x)/F(x)) where F(x) = Sum_{n>=0} x^n/n!*Product_{k=1..n} (k^2 + 1).

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PARI

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A262002 L.g.f.: log( Sum_{n>=0} x^n/n! * Product_{k=1..n} (k^2 + 1) ).

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PARI

Formula

A325050 a(n) = Product_{k=0..n} (k!^2 + 1).

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Mathematica

A228120 a(n) = (1^2 + 1)(2^2 + 1)(3^2 + 1)...(((prime(n) - 1)/2)^2 + 1).

A262001 G.f.: 1/(1 - xF'(x)/F(x)) where F(x) = Sum_{n>=0} x^n/n!Product_{k=1..n} (k^2 + 1).