A213568 Rectangular array: (row n) = b**c, where b(h) = 2^(h-1), c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

1, 4, 2, 11, 7, 3, 26, 18, 10, 4, 57, 41, 25, 13, 5, 120, 88, 56, 32, 16, 6, 247, 183, 119, 71, 39, 19, 7, 502, 374, 246, 150, 86, 46, 22, 8, 1013, 757, 501, 309, 181, 101, 53, 25, 9, 2036, 1524, 1012, 628, 372, 212, 116, 60, 28, 10, 4083, 3059, 2035, 1267

Offset: 1

Author

Clark Kimberling, Jun 18 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213569

Antidiagonal sums: A047520

Row 1, (1,3,6,...)**(1,4,9,...): A125128

Row 2, (1,3,6,...)**(4,9,16,...): A095151

Row 3, (1,3,6,...)**(9,16,25,...): A000247

Row 4, (1,3,6,...)**(16,25,36...): A208638 (?)

For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
  1...4....11...26....57....120
  2...7....18...41....88....183
  3...10...25...56....119...246
  4...13...32...71....150...309
  5...16...39...86....181...372
  6...19...46...101...212...435
		

Links

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

Crossrefs

Cf. A213500.

Programs

GAP

Flat(List([1..12], n-> List([1..n], k-> 2^(n-k+1)*(k+1) -(n+2) ))); # G. C. Greubel, Jul 26 2019

Magma

[2^(n-k+1)*(k+1) -(n+2): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 26 2019

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}]]
r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213568 *)
d = Table[t[n, n], {n, 1, 40}] (* A213569 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A047520 *)
(* Second program *)
Table[2^(n-k+1)*(k+1) -(n+2), {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 26 2019 *)

PARI

for(n=1,12, for(k=1,n, print1(2^(n-k+1)*(k+1) -(n+2), ", "))) \\ G. C. Greubel, Jul 26 2019

Sage

[[2^(n-k+1)*(k+1) -(n+2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 26 2019

Formula

T(n,k) = 4*T(n,k-1) - 5*T(n,k-2) + 2*T(n,k-3). - corrected by Clark Kimberling, Sep 03 2023

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

T(n,k) = 2^k*(n + 1) - (n + k + 1). - G. C. Greubel, Jul 26 2019