A293012 -id:A293012 - cp's OEIS frontend

A091695 Expansion of e.g.f. exp(x/(1-x)^3).

1, 1, 7, 55, 529, 6121, 82711, 1273567, 21945505, 417540529, 8680953511, 195582295591, 4742407056817, 123045795823705, 3399348471640759, 99573135106176271, 3081061456572152641, 100382623544966098657, 3433727597233037475655, 123000248740384210119319, 4603377404407810366309201

Offset: 0

Karol A. Penson, Jan 29 2004

nonn

Special values of the hypergeometric function 3F3: a(n) = n!*binomial(n+1,n-1) * hypergeom([ -n+1, 1/2*n+1, 1/2*n+3/2], [4/3, 5/3, 2], -4/27) for n>0.

Seiichi Manyama, Table of n, a(n) for n = 0..422

Column k=3 of A293012.

Cf. A082579.

CoefficientList[Series[E^(x/(1-x)^3), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)

x='x+O('x^33);
Vec(serlaplace(exp( x/(1-x)^3 )))
/* Joerg Arndt, Sep 14 2012 */

E.g.f.: exp(x/(1-x)^3).
a(n) ~ 1/2*exp(-1/27-n^(1/4)*3^(3/4)/72+sqrt(3*n)/6+4/3*n^(3/4)*3^(1/4)-n)*3^(1/8)*n^(n-1/8). - Vaclav Kotesovec, Jun 27 2013
a(n) = n! * Sum_{k=0..n} binomial(n+2*k-1,n-k)/k!. - Seiichi Manyama, Mar 06 2023

Prepended a(0)=1, Joerg Arndt, Sep 14 2012.

A291709 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} (-1)^(j-1)binomial(-k,j-1)x^j/j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 13, 24, 1, 1, 1, 5, 22, 73, 120, 1, 1, 1, 6, 33, 154, 501, 720, 1, 1, 1, 7, 46, 273, 1306, 4051, 5040, 1, 1, 1, 8, 61, 436, 2721, 12976, 37633, 40320, 1, 1, 1, 9, 78, 649, 4956, 31701, 147484, 394353, 362880, 1

Offset: 0

Seiichi Manyama, Oct 21 2017

			Square array B(j,k) begins:
   1,   1,   1,    1,    1, ...
   0,   1,   2,    3,    4, ...
   0,   1,   3,    6,   10, ...
   0,   1,   4,   10,   20, ...
   0,   1,   5,   15,   35, ...
   0,   1,   6,   21,   56, ...
Square array A(n,k) begins:
   1,   1,   1,    1,    1, ...
   1,   1,   1,    1,    1, ...
   1,   2,   3,    4,    5, ...
   1,   6,  13,   22,   33, ...
   1,  24,  73,  154,  273, ...
   1, 120, 501, 1306, 2721, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..10 give A000012, A000142, A000262, A049376, A049377, A049378, A049402, A132164, A293986, A293987, A293988.
Rows n=0-1 give A000012.
Main diagonal gives A293989.
Cf. A293012, A293991.

B[j_, k_] := (-1)^(j-1)*Binomial[-k, j-1];
A[0, ] = 1; A[n, k_] := (n-1)!*Sum[B[j, k]*A[n-j, k]/(n-j)!, {j, 1, n}];
Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2017 *)

Let B(j,k) = (-1)^(j-1)*binomial(-k,j-1) for j>0 and k>=0.

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} B(j,k)*A(n-j,k)/(n-j)! for n > 0.

A361283 Expansion of e.g.f. exp(x/(1-x)^4).

Original entry on oeis.org

1, 1, 9, 85, 961, 13041, 207001, 3746149, 75832065, 1693615681, 41302616041, 1090835399061, 30988423000129, 941461990360945, 30439632977968761, 1042973073239321701, 37731609890300935681, 1436586994020158747649

Offset: 0

Seiichi Manyama, Mar 06 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..414

Column k=4 of A293012.

Cf. A361280.

A361283 := proc(n)
    n!*add(binomial(n+3*k-1,n-k)/k!,k=0..n) ;
end proc:
seq(A361283(n),n=0..40) ; # R. J. Mathar, Mar 12 2023

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^4)))

a(n) = n!*sum(k=0, n, (-1)^(n-k)*binomial(-4*k, n-k)/k!);

a(n) = n!*sum(k=0, n, binomial(n+3*k-1, n-k)/k!);

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, (-1)^(j-1)*j*binomial(-4, j-1)*v[i-j+1]/(i-j)!)); v;

a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * binomial(-4*k,n-k)/k! = n! * Sum_{k=0..n} binomial(n+3*k-1,n-k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} (-1)^(k-1) * k * binomial(-4,k-1) * a(n-k)/(n-k)!.
D-finite with recurrence a(n) +(-5*n+4)*a(n-1) +(n-1)*(10*n-23)*a(n-2) -10*(n-1)*(n-2)*(n-3)*a(n-3) +5*(n-1)*(n-2)*(n-3)*(n-4)*a(n-4) -(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Mar 12 2023
a(n) ~ 2^(1/5) * n^(n - 1/10) * exp(-27/1280 - 13*2^(3/5)*n^(1/5)/800 + 13*2^(1/5)*n^(2/5)/240 + 2^(-6/5)*n^(3/5) + 5*2^(-8/5)*n^(4/5) - n) / sqrt(5) * (1 + 116303*2^(12/5)/(3200000*n^(1/5))). - Vaclav Kotesovec, Nov 11 2023

A361600 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)j,kj)/j!.

Original entry on oeis.org

1, 1, 2, 1, 2, 5, 1, 2, 7, 16, 1, 2, 9, 34, 65, 1, 2, 11, 58, 209, 326, 1, 2, 13, 88, 473, 1546, 1957, 1, 2, 15, 124, 881, 4626, 13327, 13700, 1, 2, 17, 166, 1457, 10526, 52537, 130922, 109601, 1, 2, 19, 214, 2225, 20326, 145867, 677594, 1441729, 986410

Offset: 0

Seiichi Manyama, Mar 17 2023

			Square array begins:
    1,    1,    1,     1,     1,     1, ...
    2,    2,    2,     2,     2,     2, ...
    5,    7,    9,    11,    13,    15, ...
   16,   34,   58,    88,   124,   166, ...
   65,  209,  473,   881,  1457,  2225, ...
  326, 1546, 4626, 10526, 20326, 35226, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A000522, A002720, A361598, A361599.
Main diagonal gives A361607.
Cf. A293012.

T(n, k) = n!*sum(j=0, n, binomial(n+(k-1)*j, k*j)/j!);

E.g.f. of column k: exp( x/(1 - x)^k ) / (1-x).

T(n,k) = Sum_{j=0..n} (n+(k-1)*j)!/(k*j)! * binomial(n,j).

A293013 a(n) = n! * [x^n] exp(x/(1 - x)^n).

Original entry on oeis.org

1, 1, 5, 55, 961, 24101, 818821, 36053515, 1984670465, 132825475081, 10583425959301, 988018789759871, 106673677280748865, 13172700275176482925, 1842428769970603518341, 289406832942160060794451, 50677793314733587473331201, 9829328870566195730521433105

Offset: 0

Ilya Gutkovskiy, Sep 28 2017

Conjecture: a(n+k) == a(n) (mod k) for all n and k. If true, then for each k, the sequence a(n) taken modulo k is a periodic sequence and the period divides k. - Peter Bala, Mar 12 2023

Seiichi Manyama, Table of n, a(n) for n = 0..274

Main diagonal of A293012. Cf. A361281.

Table[n! SeriesCoefficient[Exp[x/(1 - x)^n] , {x, 0, n}], {n, 0, 17}]
(* or *)
nmax = 20; Join[{1}, Table[n!*Sum[Binomial[(n-1)*(k+1), k*n - 1]/k!, {k, 1, n}], {n, 1, nmax}]] (* Vaclav Kotesovec, Aug 24 2025 *)

a(n) = A293012(n,n).
For n > 0, a(n) = n! * Sum_{k=1..n} binomial((n-1)*(k+1), k*n - 1)/k!. - Vaclav Kotesovec, Aug 24 2025
log(a(n)) ~ n * (2*log(n) - log(log(n)) - 1 - log(2) + log(log(n))/log(n) + (1 + 2*log(2))/(2*log(n))). - Vaclav Kotesovec, Aug 25 2025

A361277 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(k*j,n-j)/j!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 7, 1, 1, 1, 7, 19, 25, 1, 1, 1, 9, 37, 97, 81, 1, 1, 1, 11, 61, 241, 581, 331, 1, 1, 1, 13, 91, 481, 1981, 3661, 1303, 1, 1, 1, 15, 127, 841, 4881, 17551, 26335, 5937, 1, 1, 1, 17, 169, 1345, 10001, 55321, 171697, 202049, 26785, 1

Offset: 0

Seiichi Manyama, Mar 06 2023

			Square array begins:
  1,  1,   1,    1,    1,     1, ...
  1,  1,   1,    1,    1,     1, ...
  1,  3,   5,    7,    9,    11, ...
  1,  7,  19,   37,   61,    91, ...
  1, 25,  97,  241,  481,   841, ...
  1, 81, 581, 1981, 4881, 10001, ...

Columns k=0..4 give A000012, A047974, A361278, A361279, A361280.
Main diagonal gives A361281.
Cf. A293012.

T(n, k) = n!*sum(j=0, n, binomial(k*j, n-j)/j!);

E.g.f. of column k: exp(x * (1+x)^k).

T(0,k) = 1; T(n,k) = (n-1)! * Sum_{j=1..n} j * binomial(k,j-1) * T(n-j,k)/(n-j)!.

A387244 Expansion of e.g.f. exp(x^2/(1-x)^4).

Original entry on oeis.org

1, 0, 2, 24, 252, 2880, 38280, 594720, 10565520, 209502720, 4558407840, 107702179200, 2744400415680, 75016089308160, 2189152249764480, 67906418407027200, 2230160988344889600, 77271779968704921600, 2815893910009609228800, 107629691727791474841600, 4304364116456244429388800

Offset: 0

Vaclav Kotesovec, Aug 24 2025

In general, if s >= 1, 1 <= r <= s and e.g.f. = exp(x^r/(1-x)^s) then for n > 0, a(n) = n! * Sum_{k=1..n} binomial(n + (s-r)*k - 1, s*k - 1)/k!.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A293012, A000262, A082579, A091695, A361283.

Cf. A052845, A052887, A386514.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x^2/(1-x)^4))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Aug 25 2025

nmax=20; CoefficientList[Series[E^(x^2/(1-x)^4), {x, 0, nmax}], x] * Range[0, nmax]!
nmax=20; Join[{1}, Table[n!*Sum[Binomial[n+2*k-1, 4*k-1]/k!, {k, 1, n}], {n, 1, nmax}]]
Join[{1}, Table[n!*n*(n - 1)*(n + 1)/6 * HypergeometricPFQ[{1 - n/2, 3/2 - n/2, 1 + n/2, 3/2 + n/2}, {5/4, 3/2, 7/4, 2}, 1/16], {n, 1, 20}]]

For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n + 2*k - 1, 4*k - 1)/k!.
a(n) = 5*(n-1)*a(n-1) - 2*(n-1)*(5*n-11)*a(n-2) + 2*(n-2)*(n-1)*(5*n-14)*a(n-3) - 5*(n-4)*(n-3)*(n-2)*(n-1)*a(n-4) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ 2^(1/5) * 5^(-1/2) * exp(1/80 - 2^(-9/5)*n^(2/5)/3 + 5*2^(-8/5)*n^(4/5) - n) * n^(n - 1/10).

A386514 Expansion of e.g.f. exp(x^2/(1-x)^3).

Original entry on oeis.org

1, 0, 2, 18, 156, 1560, 18480, 254520, 3973200, 68947200, 1312748640, 27175024800, 607314818880, 14566195163520, 373027570755840, 10154293067318400, 292659790712889600, 8899747730037964800, 284685195814757337600, 9553060139009702515200, 335468448755976164428800

Offset: 0

Enrique Navarrete, Aug 23 2025

For n > 0, a(n) is the number of ways to linearly order n distinguishable objects into one or several lines and then choose 2 objects from each line. If the lines are also linearly ordered see A364524.

A001804(n) is the number of ways if only 1 line is used.

			a(6)=18480 since there are 10800 ways using one line, 4320 ways with 2 lines using 2 and 4 objects, 3240 ways with 2 lines of 3 objects each, and 120 ways with 3 lines of 2 objects each.

Cf. A001804, A364524.
Cf. A293012, A000262, A082579, A091695, A361283.
Cf. A052845, A052887, A387244.

nmax = 20; CoefficientList[Series[E^(x^2/(1-x)^3), {x, 0, nmax}], x] * Range[0, nmax]! (* or *)
nmax = 20; Join[{1}, Table[n!*Sum[Binomial[n + k - 1, 3*k - 1]/k!, {k, 1, n}], {n, 1, nmax}]] (* Vaclav Kotesovec, Aug 24 2025 *)

From Vaclav Kotesovec, Aug 24 2025: (Start)
For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n+k-1, 3*k-1) / k!.
a(n) = 4*(n-1)*a(n-1) - 2*(n-1)*(3*n-7)*a(n-2) + (n-2)*(n-1)*(4*n-11)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 3^(1/8) * exp(1/27 - 3^(-5/4)*n^(1/4)/8 - 3^(-1/2)*n^(1/2)/2 + 4*3^(-3/4)*n^(3/4) - n) * n^(n-1/8) / 2. (End)

A293055 a(n) = n! * [x^n] Product_{k>0} exp(binomial(n+k-1,n)*x^k).

Original entry on oeis.org

1, 1, 7, 85, 1561, 40501, 1414351, 63752137, 3580066225, 243666746281, 19695440339191, 1861672467512221, 203222602188410377, 25344097136222687005, 3576607716683440603711, 566387437351728771087121, 99916441198022855099556961, 19511402630734166295545687377

Offset: 0

Seiichi Manyama, Sep 29 2017

nonn

Cf. A293012.

Table[n!*SeriesCoefficient[Exp[x/(1-x)^(n+1)], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 29 2017 *)

a(n) = n! * [x^n] exp(x/(1-x)^(n+1)).

cp's OEIS Frontend

A091695 Expansion of e.g.f. exp(x/(1-x)^3).

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A291709 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} (-1)^(j-1)*binomial(-k,j-1)*x^j/j).

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A361283 Expansion of e.g.f. exp(x/(1-x)^4).

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Maple

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A361600 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*j,k*j)/j!.

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A293013 a(n) = n! * [x^n] exp(x/(1 - x)^n).

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A361277 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(k*j,n-j)/j!.

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A387244 Expansion of e.g.f. exp(x^2/(1-x)^4).

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Magma

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A386514 Expansion of e.g.f. exp(x^2/(1-x)^3).

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A291709 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} (-1)^(j-1)binomial(-k,j-1)x^j/j).

A361600 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)j,kj)/j!.