A039647 -id:A039647 - cp's OEIS frontend

A039692 Jabotinsky-triangle related to A039647.

1, 3, 1, 8, 9, 1, 42, 59, 18, 1, 264, 450, 215, 30, 1, 2160, 4114, 2475, 565, 45, 1, 20880, 43512, 30814, 9345, 1225, 63, 1, 236880, 528492, 420756, 154609, 27720, 2338, 84, 1, 3064320, 7235568, 6316316, 2673972, 594489, 69552, 4074, 108, 1

Offset: 1

Wolfdieter Lang

Triangle gives the nonvanishing entries of the Jabotinsky matrix for F(z)= A(z)/z = 1/(1-z-z^2) where A(z) is the g.f. of the Fibonacci numbers A000045. (Notation of F(z) as in Knuth's paper.)
E(n,x) := Sum_{m=1..n} a(n,m)*x^m, E(0,x)=1, are exponential convolution polynomials: E(n,x+y) = Sum_{k=0..n} binomial(n,k)*E(k,x)*E(n-k,y) (cf. Knuth's paper with E(n,x)= n!*F(n,x)).
E.g.f. for E(n,x): (1 - z - z^2)^(-x).
Explicit a(n,m) formula: see Knuth's paper for f(n,m) formula with f(k)= A039647(n).
E.g.f. for the m-th column sequence: ((-log(1 - z - z^2))^m)/m!.
Also the Bell transform of n!*(F(n)+F(n+2)), F(n) the Fibonacci numbers. For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

			1;
3, 1;
8, 9, 1;
42, 59, 18, 1;
264, 450, 215, 30, 1;

Vincenzo Librandi, Rows n = 1..50, flattened
D. E. Knuth, Convolution polynomials, Mathematica J. 2.1 (1992), no. 4, 67-78.
Peter Luschny, The Bell transform

Cf. A039647, A000032, A000045. Another version of this triangle is in A194938.

A000032 := proc(n) option remember; coeftayl( (2-x)/(1-x-x^2),x=0,n) ; end: A039647 := proc(n) (n-1)!*A000032(n) ; end: A039692 := proc(n,m) option remember ; if m = 1 then A039647(n) ; else add( binomial(n-1,j-1)*A039647(j)*procname(n-j,m-1),j=1..n-m+1) ; fi; end: # R. J. Mathar, Jun 01 2009

t[n_, m_] := n!*Sum[StirlingS1[k, m]*Binomial[k, n-k]*(-1)^(k+m)/k!, {k, m, n}]; Table[t[n, m], {n, 1, 9}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 21 2013, after Vladimir Kruchinin *)

T(n,m) := n!*sum((stirling1(k,m)*binomial(k,n-k))*(-1)^(k+m)/k!,k,m,n); /* Vladimir Kruchinin, Mar 26 2013 */

T(n,m) = n!*sum(k=m,n, (stirling(k,m,1)*binomial(k,n-k))*(-1)^(k+m)/k!);
for(n=1,10,for(k=1,n,print1(T(n,k),", "));print());
/* Joerg Arndt, Mar 27 2013 */

# uses[bell_matrix from A264428]
# Adds 1,0,0,0, ... as column 0 to the left side of the triangle.
bell_matrix(lambda n: factorial(n)*(fibonacci(n)+fibonacci(n+2)), 8) # Peter Luschny, Jan 16 2016

a(n, 1)= A039647(n)=(n-1)!*L(n), L(n) := A000032(n) (Lucas); a(n, m) = Sum_{j=1..n-m+1} binomial(n-1, j-1)*A039647(j)*a(n-j, m-1), n >= m >= 2.
Conjectured row sums: sum_{m=1..n} a(n,m) = A005442(n). - R. J. Mathar, Jun 01 2009
T(n,m) = n! * Sum_{k=m..n} stirling1(k,m)*binomial(k,n-k)*(-1)^(k+m)/k!. - Vladimir Kruchinin, Mar 26 2013

A331339 E.g.f.: 1 / (1 + log(1 - x - x^2)).

Original entry on oeis.org

1, 1, 5, 32, 292, 3294, 44918, 714468, 13002456, 266275200, 6060498672, 151750887936, 4145522908272, 122690391196944, 3910569680464848, 133549150323123744, 4864927063250290176, 188297220693251438208, 7716800776602560577408

Offset: 0

Ilya Gutkovskiy, Jan 14 2020

nonn

Cf. A000204, A007840, A039647.

A331339 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        add(binomial(n,k)*(k-1)!*A000204(k)*procname(n-k),k=1..n) ;
    end if;
end proc:
seq(A331339(n),n=0..42) ; # R. J. Mathar, Aug 20 2021

nmax = 18; CoefficientList[Series[1/(1 + Log[1 - x - x^2]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (k - 1)! LucasL[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k - 1)! * Lucas(k) * a(n-k).

a(n) ~ n! * 2^(n+1) * exp(n/2) / (sqrt(5*exp(1) - 4) * (sqrt(5*exp(1) - 4) - exp(1/2))^(n+1)). - Vaclav Kotesovec, Jan 26 2020

A039929 Second column of Fibonacci Jabotinsky-triangle A039692.

Original entry on oeis.org

0, 1, 9, 59, 450, 4114, 43512, 528492, 7235568, 110499696, 1862118720, 34342356960, 688092312960, 14886351037440, 345878769358080, 8590707803462400, 227153424885811200, 6371121297516595200

Offset: 1

Wolfdieter Lang

E.g.f.: (log(1-x-x^2))^2/2.

Cf. A039692, A039647, A000032, A005442.

With[{nn=20},CoefficientList[Series[Log[1-x-x^2]^2/2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 11 2017 *)

a(n) = A039692(n, 2); a(n) = (n-1)!*sum(L(j)*L(n-j)/(n-j), j=1..n-1), n >= 2, L(n)= A000032(n) (Lucas).

A328286 Expansion of e.g.f. -log(1 - x - x^2/2).

Original entry on oeis.org

1, 2, 5, 21, 114, 780, 6390, 61110, 667800, 8210160, 112152600, 1685237400, 27624920400, 490572482400, 9381882510000, 192238348302000, 4201639474032000, 97572286427616000, 2399151995223984000, 62268748888378032000, 1701213856860117600000

Offset: 1

Ilya Gutkovskiy, Oct 11 2019

nonn

Cf. A009014, A039647, A080040, A080599 (exponential transform).

b:= proc(n) b(n):= n! * (<<1|1>, <1/2|0>>^n)[1, 1] end:
a:= proc(n) option remember; `if`(n=0, 0, b(n)-add(
      binomial(n, j)*j*b(n-j)*a(j), j=1..n-1)/n)
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Oct 11 2019

nmax = 21; CoefficientList[Series[-Log[1 - x - x^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Rest
FullSimplify[Table[(n - 1)! ((1 - Sqrt[3])^n + (1 + Sqrt[3])^n)/2^n, {n, 1, 21}]]

my(x='x+O('x^25)); Vec(serlaplace(-log(1 - x - x^2/2))) \\ Michel Marcus, Oct 11 2019

a(n) = (n - 1)! * ((1 - sqrt(3))^n + (1 + sqrt(3))^n) / 2^n.

D-finite with recurrence +2*a(n) +2*(-n+1)*a(n-1) -(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Aug 20 2021

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A039692 Jabotinsky-triangle related to A039647.

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A331339 E.g.f.: 1 / (1 + log(1 - x - x^2)).

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A039929 Second column of Fibonacci Jabotinsky-triangle A039692.

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A328286 Expansion of e.g.f. -log(1 - x - x^2/2).

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