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A213550 Rectangular array:  (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution. 7
1, 5, 2, 15, 9, 3, 35, 25, 13, 4, 70, 55, 35, 17, 5, 126, 105, 75, 45, 21, 6, 210, 182, 140, 95, 55, 25, 7, 330, 294, 238, 175, 115, 65, 29, 8, 495, 450, 378, 294, 210, 135, 75, 33, 9, 715, 660, 570, 462, 350, 245, 155, 85, 37, 10, 1001, 935, 825, 690, 546 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Principal diagonal:  A002418

Antidiagonal sums:  A005585

row 1,  (1,3,6,...)**(1,2,3,...):  A000332

row 2,  (1,3,6,...)**(2,3,4,...):  A005582

row 3,  (1,3,6,...)**(3,4,5,...):  A095661

row 4,  (1,3,6,...)**(4,5,6,...):  A095667

For a guide to related arrays, see A213500.

LINKS

Table of n, a(n) for n=1..60.

FORMULA

T(n,k) = 5*T(n,k-1) - 10*T(n,k-2) + 10*T(n,k-3) - 5*T(n,k-4) + T(n,k-5).

G.f. for row n:  f(x)/g(x), where f(x) = n-(n-1)*x and g(x) = (1 - x)^5.

EXAMPLE

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

1....5....15...35....70....126

2....9....25...55....105...182

3....13...35...75....140...238

4....17...45...95....175...294

5....21...55...115...210...350

MATHEMATICA

b[n_] := n (n + 1)/2; 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}]  (* A213550 *)

d = Table[t[n, n], {n, 1, 40}] (* A002418 *)

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

s1 = Table[s[n], {n, 1, 50}] (* A005585 *)

CROSSREFS

Cf. A213500, A213548.

Sequence in context: A191722 A191435 A128142 * A283242 A246209 A297979

Adjacent sequences:  A213547 A213548 A213549 * A213551 A213552 A213553

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jun 16 2012

STATUS

approved

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Last modified January 23 19:45 EST 2020. Contains 331175 sequences. (Running on oeis4.)