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A213564 Rectangular array:  (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution. 5
1, 7, 4, 27, 21, 9, 77, 67, 43, 16, 182, 167, 127, 73, 25, 378, 357, 297, 207, 111, 36, 714, 686, 602, 467, 307, 157, 49, 1254, 1218, 1106, 917, 677, 427, 211, 64, 2079, 2034, 1890, 1638, 1302, 927, 567, 273, 81, 3289, 3234, 3054, 2730, 2282, 1757 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Principal diagonal:  A213565

Antidiagonal sums:  A101094

Row 1,  (1,3,6,...)**(1,4,9,...):  A005585

Row 2,  (1,3,6,...)**(4,9,16,...):  (k^5 +25*k^4 + 60*k^3 + 215*k^2 + 59*k)/60

Row 3,  (1,3,6,...)**(9,16,25,...):  (k^5 +35*k^4 + 30*k^3 + 505*k^2 + 149*k)/60

For a guide to related arrays, see A213500.

LINKS

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

FORMULA

T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) - T(n,k-6).

G.f. for row n:  f(x)/g(x), where f(x) = n^2 - (2*n^2 - 2n - 1)*x + ((n - 1)^2)*x^2 and g(x) = (1 - x)^6.

EXAMPLE

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

1....7.....27....77....182

4....21....67....167...357

9....43....127...297...602

16...73....207...467...917

25...111...307...677...1302

36...157...427...927...1757

MATHEMATICA

b[n_] := n (n + 1)/2; 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, 1, 60}]  (* A213564 *)

d = Table[t[n, n], {n, 1, 40}] (* A213565 *)

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

s1 = Table[s[n], {n, 1, 50}] (* A101094 *)

CROSSREFS

Cf. A213500.

Sequence in context: A085047 A213831 A282361 * A282449 A282608 A070427

Adjacent sequences:  A213561 A213562 A213563 * A213565 A213566 A213567

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jun 18 2012

STATUS

approved

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Last modified September 17 04:49 EDT 2019. Contains 327119 sequences. (Running on oeis4.)