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A053581 First differences of the Poly-Bernoulli numbers B_n^(k) with k=-2 (A027649). 5
1, 3, 10, 32, 100, 308, 940, 2852, 8620, 25988, 78220, 235172, 706540, 2121668, 6369100, 19115492, 57362860, 172121348, 516429580, 1549419812, 4648521580, 13946089028, 41839315660, 125520044132 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

Cf. A001047, A027649.

Cf. A001045.

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

Sequence in context: A104270 A038731 A244762 * A247595 A092822 A017935

Adjacent sequences:  A053578 A053579 A053580 * A053582 A053583 A053584

KEYWORD

easy,nonn

AUTHOR

Barry E. Williams, Jan 18 2000

STATUS

approved

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Last modified June 21 02:05 EDT 2021. Contains 345342 sequences. (Running on oeis4.)