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A213573 Rectangular array:  (row n) = b**c, where b(h) = 2^(h-1), c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution. 4
1, 6, 4, 21, 17, 9, 58, 50, 34, 16, 141, 125, 93, 57, 25, 318, 286, 222, 150, 86, 36, 685, 621, 493, 349, 221, 121, 49, 1434, 1306, 1050, 762, 506, 306, 162, 64, 2949, 2693, 2181, 1605, 1093, 693, 405, 209, 81, 5998, 5486, 4462, 3310, 2286, 1486 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Principal diagonal: A213574.

Antidiagonal sums: A213575.

row 1, (1,2,4,8,...)**(1,4,9,16...): A047520.

row 2, (1,2,4,8,...)**(4,9,16,25...).

row 3, (1,2,4,8,...)**(9,16,25,36...).

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

T(n,k) = 2^k*(n^2 + 2*n + 3) - (n + k + 2)^2 + 2*(n + k + 1) - 1. - G. C. Greubel, Jul 25 2019

EXAMPLE

Northwest corner (the array is read by falling antidiagonals):

   1,    6,   21,   58,  141,  318, ...

   4,   17,   50,  125,  286,  621, ...

   9,   34,   93,  222,  493, 1050, ...

  16,   57,  150,  349,  762, 1605, ...

  25,   86,  221,  506, 1093, 2286, ...

  36,  121,  306,  693, 1486, 3093, ...

  ...

MATHEMATICA

(* First program *)

b[n_]:= 2^(n-1); c[n_]:= n^2;

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, 60}] (* A213573 *)

d = Table[T[n, n], {n, 40}] (* A213574 *)

s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]

s1 = Table[s[n], {n, 1, 50}] (* A213575 *)

(* Additional programs *)

Table[2^(n-k+1)*((k+1)^2 +2)-((n+2)^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-k+1)*((k+1)^2 +2)-((n+2)^2 +2), ", "))) \\ G. C. Greubel, Jul 25 2019

(MAGMA) [2^(n-k+1)*((k+1)^2 +2)-((n+2)^2 +2): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 25 2019

(Sage) [[2^(n-k+1)*((k+1)^2 +2)-((n+2)^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-k+1)*((k+1)^2 +2)- ((n+2)^2 +2) ))); # G. C. Greubel, Jul 25 2019

CROSSREFS

Cf. A213500.

Sequence in context: A212891 A107983 A009278 * A321417 A185734 A292696

Adjacent sequences:  A213570 A213571 A213572 * A213574 A213575 A213576

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jun 18 2012

STATUS

approved

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Last modified April 21 02:10 EDT 2021. Contains 343143 sequences. (Running on oeis4.)