A342397 Expansion of the o.g.f. (2*x^2 + 3*x + 2)*x/((x + 1)^2*(x - 1)^4).

0, 2, 7, 18, 35, 62, 98, 148, 210, 290, 385, 502, 637, 798, 980, 1192, 1428, 1698, 1995, 2330, 2695, 3102, 3542, 4028, 4550, 5122, 5733, 6398, 7105, 7870, 8680, 9552, 10472, 11458, 12495, 13602, 14763, 15998, 17290, 18660, 20090, 21602, 23177, 24838, 26565, 28382, 30268, 32248, 34300, 36450

Offset

0,2

Comments

One-half of the antidiagonal sums of array A220508.

Links

Table of n, a(n) for n=0..49.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Formula

a(n) = (n+1)*(1 - (-1)^n)/16 + (7/4)*(binomial(n+3, 3) - binomial(n+2, 2)).

a(n) = (A342362(n) - (n + 1))/4.

a(2*n) = A169607(n) and a(2*n + 1) = 2*A004126(n + 1).

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 5. - Chai Wah Wu, Mar 11 2021

Mathematica

CoefficientList[Series[(2x^2+3x+2) x/((x+1)^2 (x-1)^4), {x, 0, 70}], x] (* or *) LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 2, 7, 18, 35, 62}, 70] (* Harvey P. Dale, Jul 08 2023 *)

Prog

(PARI) /* First program */
seq1(n)={my(x='x+O('x^n)); Vec((2*x^2 + 3*x + 2)*x/((x + 1)^2*(x - 1)^4), -n)}
/* Second program (array M is A220508) */
seq2(nn) = {my(M=matrix(nn+1, nn+1)); my(a=vector(nn+1)); for(n=1, nn+1, for(k=1, nn+1, M[n, k]=if(k == n, n^2-n, if(k < n, n^2-2*n+k, k^2-n)))); for(n=1, nn+1, a[n] = sum(k=1, n, M[n-k+1, k])/2); a}

Crossrefs

Cf. A004126, A169607, A220508, A342462.

Sequence in context: A141631 A172188 A077131 * A212685 A176854 A086741

Adjacent sequences: A342394 A342395 A342396 * A342398 A342399 A342400

Keyword

nonn

Author

Petros Hadjicostas, Mar 10 2021

Status

approved