A053581 First differences of the Poly-Bernoulli numbers B_n^(k) with k=-2 (A027649).

1, 3, 10, 32, 100, 308, 940, 2852, 8620, 25988, 78220, 235172, 706540, 2121668, 6369100, 19115492, 57362860, 172121348, 516429580, 1549419812, 4648521580, 13946089028, 41839315660, 125520044132

Offset: 0

Author

Barry E. Williams, Jan 18 2000

Keywords

• easy
• nonn

Comments

Also the second differences of A001047.

Equals sum of "terms added" to current row of the triangle version of A038573 to get the next row. a(3) = 32 sum of (3, 7, 7, 15) = terms appended to row 2 of the triangle in A038573. - Gary W. Adamson, Jun 04 2009

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-6).

Crossrefs

Cf. A001047, A027649.

Cf. A001045.

Cf. A038573. - Gary W. Adamson, Jun 04 2009

Programs

GAP

List([0..30], n-> 4*3^(n-1) +(0^n -3*2^n)/6) # G. C. Greubel, May 16 2019

Magma

[4*3^n/3+0^n/6-2^n/2: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

Mathematica

CoefficientList[Series[(1-x)^2/((1-2x)(1-3x)),{x,0,30}],x]  (* Harvey P. Dale, Apr 22 2011 *)

PARI

vector(30, n, n--; 4*3^(n-1) +(0^n -3*2^n)/6) \\ G. C. Greubel, May 16 2019

Sage

[4*3^(n-1) +(0^n -3*2^n)/6 for n in (0..30)] # G. C. Greubel, May 16 2019

Formula

a(n) = 5*a(n-1) - 6*a(n-2) + C(2,2-n), n>1, with a(0)=1, a(1)=3, where C(2, 2-n)=1 for n=2 and =0 for n>2.

From Paul Barry, Jun 26 2003: (Start)

Binomial transform of A000975(n+1).

G.f.: (1-x)^2/((1-2*x)*(1-3*x)).

a(n) = 4*3^n/3 + 0^n/6 - 2^n/2. (End)

a(n) = Sum_{k=0..n+1} binomial(n+1, k) * Sum_{j=0..floor(k/2)} A001045(k-2*j). - Paul Barry, Apr 17 2005

E.g.f.: (1 - 3*exp(2*x) + 8*exp(3*x))/6. - G. C. Greubel, May 16 2019