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A213750 Rectangular array:  (row n) = b**c, where b(h) = h, c(h) = 2*(n-1+h)-1, n>=1, h>=1, and ** = convolution. 5
1, 5, 3, 14, 11, 5, 30, 26, 17, 7, 55, 50, 38, 23, 9, 91, 85, 70, 50, 29, 11, 140, 133, 115, 90, 62, 35, 13, 204, 196, 175, 145, 110, 74, 41, 15, 285, 276, 252, 217, 175, 130, 86, 47, 17, 385, 375, 348, 308, 259, 205, 150, 98, 53, 19, 506, 495, 465, 420 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Principal diagonal:  A007585

Antidiagonal sums:  A002417

row 1,  (1,2,3,4,5,...)**(1,3,5,7,9,...): A000330

row 2,  (1,2,3,4,5,...)**(3,5,7,9,...): A051925

row 3,  (1,2,3,4,5,...)**(5,7,9,11,...): (2*k^3 + 15*k^2 + 13*k)/6

row 4,  (1,2,3,4,5,...)**(7,9,11,13,...): (2*k^3 + 21*k^2 + 19*k)/6

For a guide to related arrays, see A213500.

LINKS

Table of n, a(n) for n=1..59.

FORMULA

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n:  f(x)/g(x), where f(x) = (2*n - 1) - (2*n - 3)*x and g(x) = (1 - x )^4.

EXAMPLE

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

1....5....14...30....55....91

3....11...26...50....85....133

5....17...38...70....115...175

7....23...50...90....145...217

9....29...62...110...175...259

11...35...74...130...205...301

MATHEMATICA

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

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}]  (* A213750 *)

d = Table[t[n, n], {n, 1, 40}] (* A007585 *)

s1 = Table[s[n], {n, 1, 50}] (* A002417 *)

FindLinearRecurrence[s1]

FindGeneratingFunction[s1, x]

CROSSREFS

Cf. A213500.

Sequence in context: A082983 A083594 A178497 * A213774 A167583 A329029

Adjacent sequences:  A213747 A213748 A213749 * A213751 A213752 A213753

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jun 20 2012

STATUS

approved

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Last modified April 23 01:55 EDT 2021. Contains 343198 sequences. (Running on oeis4.)