A032031 -id:A032031 - cp's OEIS frontend

A331817 a(n) = (n!)^2 * Sum_{k=0..n} (2k)! / (2^k (k!)^3 * (n - k)!).

1, 2, 9, 66, 681, 9090, 148905, 2889810, 64805265, 1648535490, 46896669225, 1475099460450, 50831084252025, 1904311245686850, 77061447551313225, 3349828945512299250, 155672917524626126625, 7701743926471878533250, 404153655359180645543625

Offset: 0

Ilya Gutkovskiy, Jan 27 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..380

Cf. A000165, A001147, A032031, A034430, A052577.

[(Factorial(n))^2*&+[Factorial(2*k)/(2^k*(Factorial(k))^3*Factorial(n-k)):k in [0..n]]:n in [0..18]]; // Marius A. Burtea, Jan 27 2020

f:= gfun:-rectoproc({a(n + 2) = 2*(3 + 2*n)*a(n + 1) - 3*(n + 1)^2*a(n), a(0)=1, a(1)=2},a(n), remember):
map(f, [$0..30]); # Robert Israel, Feb 17 2020

Table[n!^2 Sum[(2 k)!/(2^k k!^3 (n - k)!), {k, 0, n}], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/Sqrt[1 - 4 x + 3 x^2], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Hypergeometric2F1[1/2, -n, 1, -2], {n, 0, 18}]

seq(n) = {Vec(serlaplace(1/(sqrt(1 - 4*x + 3*x^2 + O(x*x^n)))))} \\ Andrew Howroyd, Jan 27 2020

E.g.f.: 1 / sqrt(1 - 4*x + 3*x^2).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * (2*k - 1)!! * (n - k)!.
a(n) = n! * 2F1(1/2, -n; 1; -2).
a(n) ~ 3^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Jan 28 2020
D-finite with recurrence a(n + 2) = 2*(3 + 2*n)*a(n + 1) - 3*(n + 1)^2*a(n). - Robert Israel, Feb 17 2020

A108694 Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 2, 2, 4, 3, 6, 4, 8, 5, 10, 6, 12, . . . ] DELTA [2, 1, 4, 2, 6, 3, 8, 4, 10, 5, 12, 6, . . . ] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 2, 3, 9, 6, 13, 54, 69, 26, 75, 399, 747, 573, 150, 541, 3508, 8638, 9998, 5393, 1082, 4683, 35817, 109248, 169038, 139143, 57585, 9366, 47293, 416762, 1515531, 2935222, 3256907, 2064534, 691645, 94586

Offset: 0

Philippe Deléham, Jun 18 2005

Related to preferential arrangements of n elements (A000670) and necklaces of sets of labeled beads (A000629).

			1; 1, 2; 3, 9, 6; 13, 54, 69, 26; 75, 399, 747, 573, 150; ...

Cf. A000629, A000670, A032031, A084938.

Sum_{ k>=0 } T(n, k) = n!*3^n = A032031(n).

T(n, 0) = A000670(n); T(n, n) = A000629(n).

A153270 Triangle T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 2, read by rows.

Original entry on oeis.org

3, 3, 12, 3, 15, 105, 3, 18, 162, 1944, 3, 21, 231, 3465, 65835, 3, 24, 312, 5616, 129168, 3616704, 3, 27, 405, 8505, 229635, 7577955, 295540245, 3, 30, 510, 12240, 379440, 14418720, 648842400, 33739804800, 3, 33, 627, 16929, 592515, 25478145, 1299385395, 76663738305, 5136470466435

Offset: 0

Roger L. Bagula, Dec 22 2008

Row sums are {3, 15, 123, 2127, 69555, 3751827, 303356775, 34403458143, 5214459678387, 1018396843935195, 249088654250968899, ...}.

			Triangle begins as:
  3;
  3, 12;
  3, 15, 105;
  3, 18, 162,  1944;
  3, 21, 231,  3465,  65835;
  3, 24, 312,  5616, 129168,  3616704;
  3, 27, 405,  8505, 229635,  7577955, 295540245;
  3, 30, 510, 12240, 379440, 14418720, 648842400, 33739804800;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. this sequence (m=2), A153271 (m=3), A153272 (m=4).
Cf. A000142, A001710, A001813, A008544, A008545.
Cf. A001147, A032031, A047056, A144739, A144758.

m:=2;
function T(n,k)
  if k eq 0 then return NthPrime(m);
  else return (&*[j*n + NthPrime(m): j in [0..k]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 03 2019

m:=2; seq(seq(`if`(k=0, ithprime(m), mul(j*n + ithprime(m), j=0..k)), k=0..n), n=0..10); # G. C. Greubel, Dec 03 2019

T[n_, k_, m_]:= If[k==0, Prime[m], Product[j*n + Prime[m], {j,0,k}]];
Table[T[n,k,2], {n,0,10}, {k,0,n}]//Flatten

T(n,k) = my(m=2); if(k==0, prime(m), prod(j=0,k, j*n + prime(m)) ); \\ G. C. Greubel, Dec 03 2019

def T(n, k):
    m=2
    if (k==0): return nth_prime(m)
    else: return product(j*n + nth_prime(m) for j in (0..k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 03 2019

T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 2.

Edited by G. C. Greubel, Dec 03 2019

A196258 a(n) = 11^n*n!.

Original entry on oeis.org

1, 11, 242, 7986, 351384, 19326120, 1275523920, 98215341840, 8642950081920, 855652058110080, 94121726392108800, 11388728893445164800, 1503312213934761753600, 214973646592670930764800, 33105941575271323337779200

Offset: 0

Philippe Deléham, Oct 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000142, A000165, A032031, A047053, A052562, A047058, A051188, A051189, A051232, A051262, A145448.

[(Factorial(n)*11^n): n in [0..20]]; // Vincenzo Librandi, Oct 28 2011

Table[11^n n!,{n,0,20}] (* Harvey P. Dale, Sep 21 2015 *)

a(n)=11^n*n! \\ Charles R Greathouse IV, Nov 22 2011

a(n) = 11^n*n!.
E.g.f.: 1/(1-11*x).
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = e^(1/11).
Sum_{n>=0} (-1)^n/a(n) = e^(-1/11). (End)

A345105 a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 4, 25, 247, 3283, 54661, 1092427, 25473037, 678837319, 20351864821, 677954261635, 24842157250117, 993040102321927, 43003754679356941, 2005536858420616963, 100211634039201328381, 5341144936822423446247, 302468060262966258380773, 18136282125753572653056355

Offset: 0

Ilya Gutkovskiy, Jun 08 2021

nonn

Cf. A032031, A054687, A345102, A345104.

a[n_] := a[n] = 1 + 3 Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; A[] = 1; Do[A[x] = Normal[Integrate[3 A[x]^2 + Exp[x], x] + O[x]^(nmax + 1)], nmax]; CoefficientList[A[x], x] Range[0, nmax]!

E.g.f. A(x) satisfies: A'(x) = 3 * A(x)^2 + exp(x).

A146531 Triangle read by rows: a(n) = 3^floor(n/2)Gamma(1 + floor(n/2)); t(n,m) = a(n)/(a(n - m)a(m)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 6, 2, 6, 1, 1, 1, 2, 2, 1, 1, 1, 9, 3, 18, 3, 9, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 12, 4, 36, 6, 36, 4, 12, 1, 1, 1, 4, 4, 6, 6, 4, 4, 1, 1, 1, 15, 5, 60, 10, 90, 10, 60, 5, 15, 1

Offset: 0

Roger L. Bagula, Oct 30 2008

The row sums are: {1, 2, 5, 4, 16, 8, 44, 16, 112, 32, 272}.
The matrix inverse starts
1;
-1,1;
2,-3,1;
-2,2,-1,1;
13,-12,4,-6,1;
-13,13,-4,4,-1,1;
116,-117,39,-36,6,-9,1;
-116,116,-39,39,-6,6,-1,1;
1393,-1392,464,-468,78,-72,8,-12,1;
-1393,1393,-464,464,-78,78,-8,8,-1,1;
- R. J. Mathar, Apr 08 2013

			1;
1, 1;
1, 3, 1;
1, 1, 1, 1;
1, 6, 2, 6, 1;
1, 1, 2, 2, 1, 1;
1, 9, 3, 18, 3, 9, 1;
1, 1, 3, 3, 3, 3, 1, 1;
1, 12, 4, 36, 6, 36, 4, 12, 1;
1, 1, 4, 4, 6, 6, 4, 4, 1, 1;
1, 15, 5, 60, 10, 90, 10, 60, 5, 15,1;

A032031 := proc(n)
    3^n*n! ;
end proc:
A146531 := proc(n,m)
    A032031(floor(n/2))/A032031(floor((n-m)/2))/A032031(floor(m/2)) ;
end proc: # R. J. Mathar, Apr 08 2013

Clear[a, n, t]; a[n_] = 3^Floor[n/2]*Gamma[1 + Floor[n/2]]; t[n_, m_] = a[n]/(a[n - m]*a[m]); Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

t(n,m) = a(n)/( a(n-m)*a(m) ), where a(n) = A032031(floor(n/2)).

A190903 a(n) = Product_{k in M_n} k, M_n = {k | 1 <= k <= 3n and k mod 3 = n mod 3}.

Original entry on oeis.org

1, 1, 10, 162, 280, 12320, 524880, 1106560, 96342400, 7142567040, 17041024000, 2324549427200, 254561089305600, 664565853952000, 126757680265216000, 18763697892715776000, 52580450364682240000, 13106744139423334400000, 2480410751833883860992000

Offset: 0

Peter Luschny, Jul 03 2011

nonn

For n > 0:
a(3*n)   = A032031(3*n) = 3^(3*n) * Gamma(3*n + 1).
a(3*n-1) = A008544(3*n-1) = 3^(3*n-1) * Gamma(3*n - 1/3) / Gamma(2/3).
a(3*n+1) = A007559(3*n+1) = 3^(3*n+3/2) * Gamma(3*n + 4/3) * Gamma(2/3) / (2*Pi).

Peter Luschny, Multifactorials

Cf. A190901.

A190903 := proc(n) local k; mul(k, k = select(k-> k mod 3 = n mod 3, [$1 .. 3*n])) end: seq(A190903(n), n=0..17);

a[n_] := Switch[Mod[n, 3], 0, 3^n Gamma[n+1], 2, 3^n Gamma[n+2/3]/ Gamma[2/3], 1, 3^(n-1) Gamma[n+1/3]/Gamma[4/3]] // Round;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 25 2019 *)

a(n) = prod(k=1, 3*n, if (k % 3 == n % 3, k, 1)); \\ Michel Marcus, Jun 25 2019 and May 14 2020

From Johannes W. Meijer, Jul 04 2011: (Start)
a(3*n+3)/(a(3*n)*a(3)) = A006566(n+1); Dodecahedral numbers
a(3*n+4)/a(3*n+1) = A136214(3*n+4, 3*n+1)
a(3*n+5)/a(3*n+2) = A112333(3*n+5, 3*n+2) (End)

A068181 a(n)=-1/b(2n) where 1/(e^y-e^(y/3))= sum(i=-1,inf,b(i)*y^i).

Original entry on oeis.org

1, 18, 1944, 524880, 264539520, 214277011200, 254561089305600, 416971064282572800, 900657498850357248000, 2480410751833883860992000, 8483004771271882804592640000, 35272333838948488701496197120000

Offset: 0

Benoit Cloitre, Mar 12 2002

a(n)=A007661(6n+1)=A032031(3n+1)

A348555 Numbers k that divide the sum of the digits of 3^k * k!.

Original entry on oeis.org

1, 3, 9, 27, 72, 111, 129, 148, 161, 450, 762, 1233, 1260, 2052, 9153, 15840, 16067, 16302, 16317, 16332, 16435, 74670, 74946, 125046, 208566, 347670, 347685, 583263, 1609667, 1610942, 1616476, 1616532, 1616958, 2683143, 2700261, 4480092, 7469682, 7470432, 7492497

Offset: 1

Kevin P. Thompson, Oct 21 2021

			9 is a term because the sum of the digits of 3^9 * 9! = 7142567040 is 36 which is divisible by 9.

Cf. A032031, A007953, A052673, A108861.

Do[If[Mod[Plus @@ IntegerDigits[3^n * n!], n] == 0, Print[n]], {n, 1, 10000}]

PARI
```
isok(k) = !(sumdigits(3^k * k!) % k);
```

a(36)-a(39) from Martin Ehrenstein, Nov 19 2021

cp's OEIS Frontend

A331817 a(n) = (n!)^2 * Sum_{k=0..n} (2*k)! / (2^k * (k!)^3 * (n - k)!).

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A108694 Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 2, 2, 4, 3, 6, 4, 8, 5, 10, 6, 12, . . . ] DELTA [2, 1, 4, 2, 6, 3, 8, 4, 10, 5, 12, 6, . . . ] where DELTA is the operator defined in A084938.

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A153270 Triangle T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 2, read by rows.

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A196258 a(n) = 11^n*n!.

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A345105 a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

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A146531 Triangle read by rows: a(n) = 3^floor(n/2)*Gamma(1 + floor(n/2)); t(n,m) = a(n)/(a(n - m)*a(m)).

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A190903 a(n) = Product_{k in M_n} k, M_n = {k | 1 <= k <= 3n and k mod 3 = n mod 3}.

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Maple

Mathematica

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A068181 a(n)=-1/b(2n) where 1/(e^y-e^(y/3))= sum(i=-1,inf,b(i)*y^i).

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A331817 a(n) = (n!)^2 * Sum_{k=0..n} (2k)! / (2^k (k!)^3 * (n - k)!).

A146531 Triangle read by rows: a(n) = 3^floor(n/2)Gamma(1 + floor(n/2)); t(n,m) = a(n)/(a(n - m)a(m)).