A118395 -id:A118395 - cp's OEIS frontend

A190877 Expansion of e.g.f. exp(x+x^5).

1, 1, 1, 1, 1, 121, 721, 2521, 6721, 15121, 1844641, 20013841, 119845441, 519072841, 1816454641, 223394731561, 3501661887361, 29675906201761, 177923109591361, 844925253766561, 104750282797418881

Offset: 0

Vladimir Kruchinin, May 23 2011

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..510

Cf. A047974, A118395, A190875.

With[{nn=30},CoefficientList[Series[Exp[x+x^5],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 25 2015 *)

a(n):=n!*sum(binomial(n+(-4)*j,j)/(n+(-4)*j)!,j,0,n/4);

a(n) = if(n<5, 1, a(n-1)+5!*binomial(n-1, 4)*a(n-5)); \\ Seiichi Manyama, Feb 25 2022

a(n) = n! * Sum_{j=0..n/4} binomial(n+(-4)*j,j)/(n+(-4)*j)!.

a(n) = a(n-1) + 5! * binomial(n-1,4) * a(n-5) for n > 4. - Seiichi Manyama, Feb 25 2022

A362392 E.g.f. satisfies A(x) = exp(x + x^3 * A(x)).

Original entry on oeis.org

1, 1, 1, 7, 49, 241, 2041, 26041, 282913, 3449377, 57170161, 973059121, 16847893921, 343341027745, 7680743819113, 175958943331081, 4375517632543681, 118932887426911681, 3374685950589927649, 100735118425384221025, 3217474234925998764481

Offset: 0

Seiichi Manyama, Apr 20 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..414
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=6 of A362378.

Cf. A118395, A125500, A362393, A362430.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^3*exp(x)))))

E.g.f.: exp(x - LambertW(-x^3 * exp(x))) = -LambertW(-x^3 * exp(x))/x^3.

a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(n-2*k-1) / (k! * (n-3*k)!).

A373578 Expansion of e.g.f. exp(x * (1 + x^2)^2).

Original entry on oeis.org

1, 1, 1, 13, 49, 241, 2401, 13021, 128353, 1346689, 10615681, 140431501, 1544877841, 17576665393, 264566466529, 3226728670621, 48376006929601, 766753039205761, 11052669865900033, 197019825098096269, 3271213100827557361, 56597110823949654001

Offset: 0

Seiichi Manyama, Jun 10 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A118395, A190863, A373577.

Cf. A361278.

nmax = 20; CoefficientList[Series[E^(x*(1 + x^2)^2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 11 2024 *)

a(n) = n!*sum(k=0, 2*n\5, binomial(2*n-4*k, k)/(n-2*k)!);

a(n) = n! * Sum_{k=0..floor(2*n/5)} binomial(2*n-4*k,k)/(n-2*k)!.
a(n) == 1 (mod 12).
a(n) = a(n-1) + 6*(n-1)*(n-2)*a(n-3) + 5*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5).
a(n) ~ 5^(n/5 - 1/2) * exp(7*5^(-11/5)*n^(1/5) + 2*5^(-3/5)*n^(3/5) - 4*n/5) * n^(4*n/5). - Vaclav Kotesovec, Jun 11 2024

A246607 Expansion of e.g.f. exp(x - x^3).

Original entry on oeis.org

1, 1, 1, -5, -23, -59, 241, 2311, 9745, -30743, -529919, -3161069, 6984121, 216832045, 1696212337, -2117117729, -138721306079, -1359994188719, 367573878145, 127713732858667, 1523067770484361, 1104033549399061, -159815269852521359, -2270787199743845705, -3946710127731620303

Offset: 0

Robert G. Wilson v, Aug 31 2014

sign

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

Cf. A118395, A293604.

Range[0, 24]! CoefficientList[Series[Exp[x - x^3], {x, 0, 24}], x] (* Robert G. Wilson v, Aug 31 2014, with correction from Vincenzo Librandi *)

default(seriesprecision, 30); serlaplace(exp(x-x^3)) \\ Michel Marcus, Aug 31 2014

a(n) = n!*sum(k=0, n\3, (-1)^k*binomial(n-2*k, k)/(n-2*k)!); \\ Seiichi Manyama, Feb 25 2022

a(n) = if(n<3, 1, a(n-1)-3!*binomial(n-1, 2)*a(n-3)); \\ Seiichi Manyama, Feb 25 2022

From Seiichi Manyama, Feb 25 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k * binomial(n-2*k,k)/(n-2*k)!.
a(n) = a(n-1) - 3! * binomial(n-1,2) * a(n-3) for n > 2. (End)

A118396 Eigenvector of triangle A118394; E.g.f.: exp( Sum_{n>=0} x^(3^n) ).

Original entry on oeis.org

1, 1, 1, 7, 25, 61, 481, 2731, 10417, 454105, 4309921, 23452111, 592433161, 6789801877, 46254009985, 893881991731, 11548704851041, 93501748795441, 4828847934591937, 83867376656907415, 823025819684123641, 33409213329178701421, 640457721676922946721

Offset: 0

Paul D. Hanna, May 07 2006

nonn

E.g.f. of triangle A118394 is: exp(x+y*x^3), where A118394(n,k) = n!/k!/(n-3*k)!. More generally, given a triangle with e.g.f.: exp(x+y*x^b), the eigenvector will have e.g.f.: exp( Sum_{n>=0} x^(b^n) ).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A118394, A118395; variants: A118393, A118932.

a:= proc(n) option remember; `if`(n=0, 1, add((j-> j!*
      a(n-j)*binomial(n-1, j-1))(3^i), i=0..ilog[3](n)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 01 2017

a[n_] := a[n] = If[n==0, 1, Sum[Function[j, j! a[n-j] Binomial[n-1, j-1]][3^i], {i, 0, Log[3, n]}]];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 07 2020, after Alois P. Heinz *)

{a(n)=n!*polcoeff(exp(sum(k=0,ceil(log(n+1)/log(3)),x^(3^k))+x*O(x^n)),n)}

a(n) = Sum_{k=0..[n/3]} n!/k!/(n-3*k)! *a(k) for n>=0, with a(0)=1.

A118393 Eigenvector of triangle A059344. E.g.f.: exp( Sum_{n>=0} x^(2^n) ).

Original entry on oeis.org

1, 1, 3, 7, 49, 201, 1411, 7183, 108417, 816049, 9966691, 80843511, 1381416433, 14049020857, 216003063459, 2309595457471, 72927332784001, 1046829280528353, 23403341433961027, 329565129021010279, 9695176730057249841, 160632514329660035881

Offset: 0

Paul D. Hanna, May 07 2006

nonn

E.g.f. of A059344 is: exp(x+y*x^2). More generally, given a triangle with e.g.f.: exp(x+y*x^b), the eigenvector will have e.g.f.: exp( Sum_{n>=0} x^(b^n) ).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A059344, variants: A118395, A118930.

function a(n)
  if n eq 0 then return 1;
  else return (&+[ (Factorial(n)/(Factorial(k)*Factorial(n-2*k)))*a(k): k in [0..Floor(n/2)]]);
  end if; return a; end function;
[a(n): n in [0..25]]; // G. C. Greubel, Feb 18 2021

A118393 := proc(n)
    option remember;
    if n <=1 then
        1;
    else
        n!*add(procname(k)/k!/(n-2*k)!,k=0..n/2) ;
    end if;
end proc:
seq(A118393(n),n=0..20) ; # R. J. Mathar, Aug 19 2014
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add((j-> j!*
      a(n-j)*binomial(n-1, j-1))(2^i), i=0..ilog2(n)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 01 2017

a[0] = 1; a[n_] := a[n] = Sum[n!/k!/(n - 2*k)!*a[k], {k, 0, n/2}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 18 2018 *)

a(n)=n!*polcoeff(exp(sum(k=0,#binary(n),x^(2^k))+x*O(x^n)),n)

f=factorial;
def a(n): return 1 if n==0 else sum((f(n)/(f(k)*f(n-2*k)))*a(k) for k in (0..n//2))
[a(n) for n in (0..25)] # G. C. Greubel, Feb 18 2021

a(n) = Sum_{k=0..[n/2]} n!/k!/(n-2*k)! *a(k) for n>=0, with a(0)=1.

A118394 Triangle T(n,k) = n!/(k!(n-3k)!), for n >= 3*k >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 24, 1, 60, 1, 120, 360, 1, 210, 2520, 1, 336, 10080, 1, 504, 30240, 60480, 1, 720, 75600, 604800, 1, 990, 166320, 3326400, 1, 1320, 332640, 13305600, 19958400, 1, 1716, 617760, 43243200, 259459200, 1, 2184, 1081080, 121080960, 1816214400

Offset: 0

Paul D. Hanna, May 07 2006

Row sums form A118395.

Eigenvector is A118396.

			Triangle begins:
  1;
  1;
  1;
  1,    6;
  1,   24;
  1,   60;
  1,  120,    360;
  1,  210,   2520;
  1,  336,  10080;
  1,  504,  30240,    60480;
  1,  720,  75600,   604800;
  1,  990, 166320,  3326400;
  1, 1320, 332640, 13305600, 19958400;
  ...

G. C. Greubel, Rows n = 0..150 of the triangle, flattened

Cf. A118395 (row sums), A118396 (eigenvector).

Variants: A059344, A118931.

F:= Factorial;
[F(n)/(F(k)*F(n-3*k)): k in [0..Floor(n/3)], n in [0..20]]; // G. C. Greubel, Mar 07 2021

T[n_, k_] := n!/(k!(n-3k)!);
Table[T[n, k], {n, 0, 14}, {k, 0, Floor[n/3]}] // Flatten (* Jean-François Alcover, Nov 04 2020 *)

T(n,k)=if(n<3*k || k<0,0,n!/k!/(n-3*k)!)

f=factorial;
flatten([[f(n)/(f(k)*f(n-3*k)) for k in [0..n/3]] for n in [0..20]]) # G. C. Greubel, Mar 07 2021

E.g.f.: A(x,y) = exp(x + y*x^3).

A358560 a(n) = Sum_{k=0..floor(n/3)} (n-k)!/(k! * (n-3*k)!).

Original entry on oeis.org

1, 1, 1, 3, 7, 13, 33, 91, 223, 597, 1753, 4963, 14391, 44413, 137137, 427083, 1382383, 4534981, 14981673, 50719507, 174494983, 605276973, 2135204161, 7647369403, 27643067007, 101211363253, 375548195833, 1406858084931, 5326762882903, 20403498329437

Offset: 0

Seiichi Manyama, Nov 22 2022

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000930, A118395, A358547.

a(n) = sum(k=0, n\3, (n-k)!/(k!*(n-3*k)!));

a(n) = (4 * a(n-1) - a(n-2) + 2 * (2*n-3) * a(n-3))/3 for n > 2.

a(n) ~ c * 2^(2*n/3) * n^(n/3) / (3^(n/3) * exp(n/3 - 2^(1/3) * n^(2/3) / 3^(2/3) + n^(1/3) / (2^(4/3) * 3^(7/3)))) * (1 + 7795/(5832*6^(2/3)*n^(1/3)) + 135724109/(2040733440*6^(1/3)*n^(2/3)) - 5962064767253/(42845606719488*n)), where c = 0.46562048925..., conjecture: c = sqrt(2) * exp(-1/81) / 3. - Vaclav Kotesovec, Nov 25 2022

A373577 Expansion of e.g.f. exp(x * (1 + x^2)^(3/2)).

Original entry on oeis.org

1, 1, 1, 10, 37, 136, 1261, 6616, 45865, 479872, 3206521, 31165696, 356045581, 3082798720, 37528974757, 443190912256, 4792765859281, 69943918698496, 875123733523825, 11059833224507392, 179428023035501941, 2557848382674927616, 37699048392962570461

Offset: 0

Seiichi Manyama, Jun 10 2024

nonn

Cf. A118395, A190863, A373578.

a(n) = n!*sum(k=0, n\2, binomial(3*n/2-3*k, k)/(n-2*k)!);

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(3*n/2-3*k,k)/(n-2*k)!.

a(n) == 1 mod 9.

A376564 E.g.f. satisfies A(x) = exp( xA(x) (1 + x^2*A(x)^2) ).

Original entry on oeis.org

1, 1, 3, 22, 245, 3456, 60487, 1283584, 31971753, 912448000, 29369155211, 1053204332544, 41646891006877, 1800306963331072, 84464613778359375, 4274750510822588416, 232146299393990454353, 13465725621588464173056, 830921722002492358973203

Offset: 0

Seiichi Manyama, Sep 28 2024

nonn

Index entries for reversions of series

Cf. A088695, A376565.

Cf. A118395, A376476.

a(n) = n!*sum(k=0, n\3, (n+1)^(n-2*k-1)*binomial(n-2*k, k)/(n-2*k)!);

E.g.f.: (1/x) * Series_Reversion( x*exp(-x * (1 + x^2)) ).

a(n) = n! * Sum_{k=0..floor(n/3)} (n+1)^(n-2*k-1) * binomial(n-2*k,k)/(n-2*k)!.

cp's OEIS Frontend

A190877 Expansion of e.g.f. exp(x+x^5).

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A362392 E.g.f. satisfies A(x) = exp(x + x^3 * A(x)).

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A373578 Expansion of e.g.f. exp(x * (1 + x^2)^2).

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A246607 Expansion of e.g.f. exp(x - x^3).

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A118396 Eigenvector of triangle A118394; E.g.f.: exp( Sum_{n>=0} x^(3^n) ).

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A118393 Eigenvector of triangle A059344. E.g.f.: exp( Sum_{n>=0} x^(2^n) ).

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A118394 Triangle T(n,k) = n!/(k!*(n-3*k)!), for n >= 3*k >= 0, read by rows.

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A118394 Triangle T(n,k) = n!/(k!(n-3k)!), for n >= 3*k >= 0, read by rows.

A376564 E.g.f. satisfies A(x) = exp( xA(x) (1 + x^2*A(x)^2) ).