A001779 Expansion of 1/((1+x)(1-x)^8).

1, 7, 29, 91, 239, 553, 1163, 2269, 4166, 7274, 12174, 19650, 30738, 46782, 69498, 101046, 144111, 201993, 278707, 379093, 508937, 675103, 885677, 1150123, 1479452, 1886404, 2385644, 2993972

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is the number of positive terms in the expansion of (a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 - z)^n. Also the convolution of A001769 and A000012; A001753 and A001477; A001752 and A000217; A002623 and A000292; A002620 and A000332; A004526 and A000389. - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Feb 13 2007

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (7,-20,28,-14,-14,28,-20,7,-1).

Crossrefs

Cf. A001769 (first differences), A169795 (binomial transf.)

Programs

Magma

[1/80640*(2*n+9) *(4*n^6 +108*n^5 +1138*n^4 +5904*n^3 +15628*n^2 +19638*n +8925)+(-1)^n/256 : n in [0..30]]; // Vincenzo Librandi, Oct 08 2011

Maple

A001779 := proc(n) 1/80640*(2*n+9) *(4*n^6 +108*n^5 +1138*n^4 +5904*n^3 +15628*n^2 +19638*n +8925)+(-1)^n/256 ; end proc:
seq(A001779(n),n=0..50) ; # R. J. Mathar, Mar 22 2011

Mathematica

CoefficientList[Series[1/((1 + x) (1 - x)^8), {x, 0, 50}], x] (* G. C. Greubel, Nov 24 2017 *)
LinearRecurrence[{7,-20,28,-14,-14,28,-20,7,-1},{1,7,29,91,239,553,1163,2269,4166},30] (* Harvey P. Dale, Jan 21 2023 *)

PARI

a(n)=(2*n+9)*(4*n^6+108*n^5+1138*n^4+5904*n^3+15628*n^2+19638*n + 8925)/80640 +(-1)^n/256 \\ Charles R Greathouse IV, Apr 17 2012

Formula

a(n) = (-1)^{7-n}*Sum_{i=0..n} ((-1)^(7-i)*binomial(7+i,i)). - Sergio Falcon, Feb 13 2007

a(n)+a(n+1) = A000580(n+8). - R. J. Mathar, Jan 06 2021