A213582 Rectangular array: (row n) = b**c, where b(h) = -1 + 2^h, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

1, 5, 2, 16, 9, 3, 42, 27, 13, 4, 99, 68, 38, 17, 5, 219, 156, 94, 49, 21, 6, 466, 339, 213, 120, 60, 25, 7, 968, 713, 459, 270, 146, 71, 29, 8, 1981, 1470, 960, 579, 327, 172, 82, 33, 9, 4017, 2994, 1972, 1207, 699, 384, 198, 93, 37, 10, 8100, 6053, 4007, 2474, 1454, 819, 441, 224, 104, 41, 11

Offset

1,2

Comments

Principal diagonal: A213583.

Antidiagonal sums: A156928.

Row 1, (1,3,7,15,31,...)**(1,2,3,4,5,...): A002662.

Row 2, (1,3,7,15,31,...)**(2,3,4,5,6,...)

Row 3, (1,3,7,15,31,...)**(3,4,5,6,7,...)

For a guide to related arrays, see A213500.

Links

Clark Kimberling, Antidiagonals n = 1..60, flattened

Formula

T(n,k) = 5*T(n,k-1) - 9*T(n,k-2) + 7*T(n,k-3) - 2*T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = n - (n-1)*x and g(x) = (1-2*x) *(1-x)^3.

T(n,k) = 2*(n+1)*(2^k - 1) - k*(k + 2*n + 3)/2. - G. C. Greubel, Jul 08 2019

Example

Northwest corner (the array is read by falling antidiagonals):
1...5....16...42....99....219
2...9....27...68....156...339
3...13...38...94....213...459
4...17...49...120...270...579
5...21...60...146...327...699
6...25...71...172...384...819

Mathematica

(* First program *)
b[n_]:= 2^n - 1; c[n_]:= n;
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213582 *)
r[n_]:= Table[T[n, k], {k, 40}]
Table[T[n, n], {n, 1, 40}] (* A213583 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A156928 *)
(* Second program *)
Table[2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 08 2019 *)

Prog

(PARI) t(n, k) = 2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2;
for(n=1, 12, for(k=1, n, print1(t(n, k), ", "))) \\ G. C. Greubel, Jul 08 2019
(Magma) [[2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2: k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 08 2019
(Sage) [[2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2 for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 08 2019
(GAP) Flat(List([1..12], n-> List([1..n], k-> 2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2 ))) # G. C. Greubel, Jul 08 2019

Crossrefs

Cf. A213500, A213571.

Sequence in context: A298630 A248258 A279150 * A364108 A367290 A111267

Adjacent sequences: A213579 A213580 A213581 * A213583 A213584 A213585

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Jun 19 2012

Status

approved