A213571 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.

1, 5, 3, 16, 13, 7, 42, 38, 29, 15, 99, 94, 82, 61, 31, 219, 213, 198, 170, 125, 63, 466, 459, 441, 406, 346, 253, 127, 968, 960, 939, 897, 822, 698, 509, 255, 1981, 1972, 1948, 1899, 1809, 1654, 1402, 1021, 511, 4017, 4007, 3980, 3924, 3819, 3633

Offset

1,2

Comments

Principal diagonal: A213572.

Antidiagonal sums: A213581.

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

Row 2, (1,2,3,4,5,...)**(3,7,15,31,63,...).

Row 3, (1,2,3,4,5,...)**(7,15,31,63,...).

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) = x*(-1 + 2^n - (-2 + 2^n)*x) and g(x) = (1 - 2*x)(1 - x)^3.

T(n,k) = 2^(n+k+1) - 2^n*(k+2) - binomial(k+1, 2). - G. C. Greubel, Jul 25 2019

Example

Northwest corner (the array is read by falling antidiagonals):
   1,    5,   16,   42,   99,  219, ...
   3,   13,   38,   94,  213,  459, ...
   7,   29,   82,  198,  441,  939, ...
  15,   61,  170,  406,  897, 1899, ...
  31,  125,  346,  822, 1809, 3819, ...
  ...

Mathematica

(* First program *)
b[n_]:= n; c[n_]:= -1 + 2^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}]]
r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213571 *)
d = Table[t[n, n], {n, 1, 40}] (* A213572 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213581 *)
(* Additional programs *)
Table[2^(n+2) -2^k*(n-k+3) -Binomial[n-k+2, 2], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 25 2019 *)

Prog

(PARI) for(n=1, 12, for(k=1, n, print1(2^(n+2) -2^k*(n-k+3) -binomial(n-k+2, 2), ", "))) \\ G. C. Greubel, Jul 25 2019
(Magma) [2^(n+2) -2^k*(n-k+3) -Binomial(n-k+2, 2): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 25 2019
(Sage) [[2^(n+2) -2^k*(n-k+3) -binomial(n-k+2, 2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 25 2019
(GAP) Flat(List([1..12], n-> List([1..n], k-> 2^(n+2) -2^k*(n-k+3) -Binomial(n-k+2, 2) ))); # G. C. Greubel, Jul 25 2019

Crossrefs

Cf. A213500, A213587.

Sequence in context: A053371 A199005 A217415 * A185880 A292243 A105201

Adjacent sequences: A213568 A213569 A213570 * A213572 A213573 A213574

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Jun 19 2012

Status

approved