login
First differences of the Poly-Bernoulli numbers B_n^(k) with k=-2 (A027649).
5

%I #35 Sep 08 2022 08:45:00

%S 1,3,10,32,100,308,940,2852,8620,25988,78220,235172,706540,2121668,

%T 6369100,19115492,57362860,172121348,516429580,1549419812,4648521580,

%U 13946089028,41839315660,125520044132

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

%C Also the second differences of A001047.

%C 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

%H Vincenzo Librandi, <a href="/A053581/b053581.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6).

%F 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.

%F From _Paul Barry_, Jun 26 2003: (Start)

%F Binomial transform of A000975(n+1).

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

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

%F 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

%F E.g.f.: (1 - 3*exp(2*x) + 8*exp(3*x))/6. - _G. C. Greubel_, May 16 2019

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

%o (Magma) [4*3^n/3+0^n/6-2^n/2: n in [0..30]]; // _Vincenzo Librandi_, Jul 17 2011

%o (PARI) vector(30, n, n--; 4*3^(n-1) +(0^n -3*2^n)/6) \\ _G. C. Greubel_, May 16 2019

%o (Sage) [4*3^(n-1) +(0^n -3*2^n)/6 for n in (0..30)] # _G. C. Greubel_, May 16 2019

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

%Y Cf. A001047, A027649.

%Y Cf. A001045.

%Y Cf. A038573. - _Gary W. Adamson_, Jun 04 2009

%K easy,nonn

%O 0,2

%A _Barry E. Williams_, Jan 18 2000