A118589 -id:A118589 - cp's OEIS frontend

A293669 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(Sum_{j=1..k} x^j).

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 7, 1, 1, 1, 3, 13, 25, 1, 1, 1, 3, 13, 49, 81, 1, 1, 1, 3, 13, 73, 261, 331, 1, 1, 1, 3, 13, 73, 381, 1531, 1303, 1, 1, 1, 3, 13, 73, 501, 2611, 9073, 5937, 1, 1, 1, 3, 13, 73, 501, 3331, 19993, 63393, 26785, 1, 1, 1, 3, 13, 73, 501, 4051, 27553, 165873, 465769, 133651, 1

Offset: 0

Seiichi Manyama, Oct 14 2017

			Square array begins:
   1,  1,   1,   1,   1, ...
   1,  1,   1,   1,   1, ...
   1,  3,   3,   3,   3, ...
   1,  7,  13,  13,  13, ...
   1, 25,  49,  73,  73, ...
   1, 81, 261, 381, 501, ...

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

Columns k=1..9 give A000012, A047974, A118589, A193930, A193931, A193932, A193933, A306623, A306624.
Rows n=0-1 give A000012.
Main diagonal gives A000262.
Cf. A293718, A293724.

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(n-j, k)*binomial(n-1, j-1)*j!, j=1..min(n, k)))
    end:
seq(seq(A(n, 1+d-n), n=0..d), d=0..12);  # Alois P. Heinz, Nov 11 2020

A[0, ] = 1; A[n /; n >= 0, k_ /; k >= 1] := A[n, k] = (n-1)!*Sum[j*A[n-j, k]/(n-j)!, {j, 1, Min[k, n]}]; A[, ] = 0;
Table[A[n, d-n+1], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 13 2021 *)

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

A294250 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} (1+x^j) - 1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 3, 13, 1, 0, 1, 1, 3, 19, 49, 1, 0, 1, 1, 3, 19, 97, 261, 1, 0, 1, 1, 3, 19, 121, 681, 1531, 1, 0, 1, 1, 3, 19, 121, 921, 5971, 9073, 1, 0, 1, 1, 3, 19, 121, 1041, 8491, 50443, 63393, 1, 0, 1, 1, 3, 19, 121, 1041

Offset: 0

Seiichi Manyama, Oct 26 2017

			Square array A(n,k) begins:
   1, 1,   1,   1,   1, ...
   0, 1,   1,   1,   1, ...
   0, 1,   3,   3,   3, ...
   0, 1,  13,  19,  19, ...
   0, 1,  49,  97, 121, ...
   0, 1, 261, 681, 921, ...

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

Columns k=0..5 give A000007, A000012, A118589, A294251, A294252, A294253.
Rows n=0 gives A000012.
Main diagonal gives A293840.
Cf. A070936, A294212, A294254.

B(j,k) is the coefficient of Product_{i=1..k} (1+x^i).

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(A000217(k),n)} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0.

A193930 E.g.f.: exp(x+x^2+x^3+x^4).

Original entry on oeis.org

1, 1, 3, 13, 73, 381, 2611, 19993, 165873, 1436473, 14004451, 145099461, 1584090553, 18196817653, 223416271443, 2865429498961, 38330181602401, 535448870264433, 7823019065848003, 118402856414023933, 1856454825152993961, 30160691907215561581

Offset: 0

Vladimir Kruchinin, Aug 09 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..524

Cf. A047974, A118589.

I:=[1,1,3,13]; [n le 4 select I[n] else Self(n-1)+2*(n-2)*Self(n-2)+3*(n-2)*(n-3)*Self(n-3)+4*(n-2)*(n-3)*(n-4)*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Oct 12 2014

A193930:=n->`if`(n=0,1,n!*add(add( (-1)^i*binomial(k,k-i)*binomial(n-4*i-1,k-1)/k!, i=0..(n-k)/4), k=1..n)): seq(A193930(n), n=0..20); # Wesley Ivan Hurt, Oct 11 2014
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=1..min(n, 4)))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 29 2017

RecurrenceTable[{a[n] == a[n - 1] + 2 (n - 1) a[n - 2] + (n - 1) (n - 2) 3 a[n - 3] + (n - 1) (n - 2) (n - 3) 4 a[n - 4], a[0] == 1,
a[1] == 1, a[2] == 3, a[3] == 13}, a, {n, 0, 20}] (* Geoffrey Critzer, Oct 11 2014 *)
With[{nn=30},CoefficientList[Series[Exp[x+x^2+x^3+x^4],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 20 2021 *)

a(n):=if n=0 then 1 else n!*sum(sum((-1)^i*binomial(k,k-i)*binomial(n-4*i-1,k-1),i,0,(n-k)/4)/k!,k,1,n);

x = y + O(y^50); concat(0, Vec(serlaplace(exp(x+x^2+x^3+x^4)))) \\ Michel Marcus, Oct 12 2014

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

a(n) = a(n-1) + 2*(n-1)*a(n-2) + 3*(n-1)(n-2)*a(n-3) + 4*(n-1)*(n-2)*(n-3)*a(n-4). - Geoffrey Critzer, Oct 11 2014

A334562 E.g.f.: exp(-(x + x^2 + x^3)).

Original entry on oeis.org

1, -1, -1, -1, 25, 19, -209, -2269, 2801, 68615, 371071, -2499641, -28306871, -58645861, 1964456495, 15133179179, -37119981599, -1861550428529, -9225044407169, 110317002942095, 2150185424201081, 3953685082287779, -233260896605772881, -2920858244957587661, 7649165533910291665

Offset: 0

Seiichi Manyama, May 06 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..500

Column 3 of A334561.

Cf. A118589, A334569.

With[{nn=30},CoefficientList[Series[Exp[-(x+x^2+x^3)],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Nov 26 2020 *)

N=40; x='x+O('x^N); Vec(serlaplace(exp(-x-x^2-x^3)))

a(0) = 1 and a(n) = - (n-1)! * Sum_{k=1..min(3,n)} k*a(n-k)/(n-k)!.

D-finite with recurrence a(n) + a(n-1) +2*(n-1)*a(n-2) +3*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, May 07 2020

A118588 Triangle generated by e.g.f.: A(x,y) = exp(x + y*(x^2+x^3)), read by rows of length [n/2+1].

Original entry on oeis.org

1, 1, 1, 2, 1, 12, 1, 36, 12, 1, 80, 180, 1, 150, 1260, 120, 1, 252, 5460, 3360, 1, 392, 17640, 43680, 1680, 1, 576, 46872, 342720, 75600, 1, 810, 108360, 1839600, 1587600, 30240, 1, 1100, 225720, 7539840, 20235600, 1995840, 1, 1452, 433620, 25391520

Offset: 0

Paul D. Hanna, May 08 2006

E.g.f. V(x) of eigenvector A119013 satisfies: V(x) = exp(x)*V(x^2+x^3); note the similarity to e.g.f. of this triangle.

			Triangle begins:
1;
1;
1,2;
1,12;
1,36,12;
1,80,180;
1,150,1260,120;
1,252,5460,3360;
1,392,17640,43680,1680;
1,576,46872,342720,75600; ...
O.g.f. for columns:
0!/0!*(1)/(1-x);
2!/1!*(1+2*x)/(1-x)^4;
4!/2!*(1+8*x+21*x^2)/(1-x)^7;
6!/3!*(1+18*x+129*x^2+356*x^3)/(1-x)^10;
8!/4!*(1+32*x+438*x^2+2984*x^3+8425*x^4)/(1-x)^13; ...

Cf. A118589 (row sums), A119013 (eigenvector).

{T(n,k)=n!*polcoeff(polcoeff(exp(x+y*(x^2+x^3)+x*O(x^n)+y*O(y^k)),n,x),k,y)}

A119013 Eigenvector of triangle A118588; E.g.f. satisfies: A(x) = exp(x)*A(x^2+x^3).

Original entry on oeis.org

1, 1, 3, 13, 73, 621, 5491, 60313, 743793, 10115353, 158914531, 2815311621, 55094081593, 1142894689093, 25142695616403, 594557634923281, 15084112106943841, 407999468524242993, 11669035487641120963

Offset: 0

Paul D. Hanna, May 08 2006

nonn

E.g.f. of triangle A118588 is exp(x + y*(x^2+x^3)); note the similarity to the e.g.f. of this sequence. More generally, the e.g.f. of an eigenvectors can be determined from the e.g.f. of a triangle as follows. [ Given a triangle with e.g.f.: exp(x + y*x*F(x)) such that F(0) = 0, then the eigenvector has e.g.f.: exp(G(x)) where o.g.f. G(x) satisfies: G(x) = x + G(x*F(x)). ]

			A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 73*x^4/4! + 621*x^5/5! +...
log(A(x)) = o.g.f. of A014535 =
x + x^2+ x^3+ x^4+ 2*x^5+ 2*x^6+ 3*x^7+ 4*x^8+ 5*x^9+ 8*x^10 +...

Cf. A118588 (triangle), A118589 (row sums), A014535 (log(A(x))).

{a(n)=if(n==0,1,sum(k=0,n\2,a(k)*n!*polcoeff(polcoeff(exp(x+y*(x^2+x^3)+x*O(x^n)+y*O(y^k)),n,x),k,y)))}

Log(A(x)) = o.g.f. of A014535 (B-trees of order 3 with n leaves).

A376512 Expansion of e.g.f. exp(x^2 * (1 + x)).

Original entry on oeis.org

1, 0, 2, 6, 12, 120, 480, 2520, 21840, 120960, 937440, 8316000, 60540480, 570810240, 5465940480, 49037788800, 523588665600, 5504686387200, 57816850291200, 678823104960000, 7844848544332800, 93064133530368000, 1184800751111577600, 14967781957781452800

Offset: 0

Seiichi Manyama, Sep 25 2024

Cf. A047974, A376513.

Cf. A118589, A227937, A373740.

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

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(k,n-2*k)/k!.
a(n) = (n-1) * (2*a(n-2) + 3*(n-2)*a(n-3)).
a(n) ~ 3^(n/3 - 1/2) * exp(4/81 - 2*3^(-7/3)*n^(1/3) + 3^(-2/3)*n^(2/3) - 2*n/3) * n^(2*n/3) * (1 + 223/(3^(20/3)*n^(1/3))). - Vaclav Kotesovec, Sep 26 2024

cp's OEIS Frontend

A293669 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(Sum_{j=1..k} x^j).

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A294250 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} (1+x^j) - 1).

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A193930 E.g.f.: exp(x+x^2+x^3+x^4).

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A334562 E.g.f.: exp(-(x + x^2 + x^3)).

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A118588 Triangle generated by e.g.f.: A(x,y) = exp(x + y*(x^2+x^3)), read by rows of length [n/2+1].

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A119013 Eigenvector of triangle A118588; E.g.f. satisfies: A(x) = exp(x)*A(x^2+x^3).

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A376512 Expansion of e.g.f. exp(x^2 * (1 + x)).

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