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A213781 Rectangular array:  (row n) = b**c, where b(h) = 1+[h/2], c(h) = n-1+h, n>=1, h>=1, [ ] = floor, and ** = convolution. 4
1, 4, 2, 9, 7, 3, 17, 14, 10, 4, 28, 25, 19, 13, 5, 43, 39, 33, 24, 16, 6, 62, 58, 50, 41, 29, 19, 7, 86, 81, 73, 61, 49, 34, 22, 8, 115, 110, 100, 88, 72, 57, 39, 25, 9, 150, 144, 134, 119, 103, 83, 65, 44, 28, 10, 191, 185, 173, 158, 138, 118, 94, 73, 49, 31 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Principal diagonal: A213782.

Antidiagonal sums: A005712.

row 1,  (1,2,2,3,3,4,4,...)**(1,2,3,4,5,6,7,...): A005744.

row 2,  (1,2,2,3,3,4,4,...)**(2,3,4,5,6,7,8,...).

row 3,  (1,2,2,3,3,4,4,...)**(3,4,5,6,7,8,9,...).

For a guide to related arrays, see A213500.

LINKS

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

FORMULA

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

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

EXAMPLE

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

1...4....9....17...28...43....62

2...7....14...25...39...58....81

3...10...19...33...50...73....100

4...13...24...41...61...88....119

5...16...29...49...72...103...138

6...19...34...57...83...118...157

7...22...39...65...94...133...176

MATHEMATICA

b[n_] := Floor[(n + 2)/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}]  (* A213781 *)

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

s1 = Table[s[n], {n, 1, 50}] (* A005712 *)

CROSSREFS

Cf. A213500, A213778.

Sequence in context: A257734 A257732 A101690 * A257730 A246378 A260422

Adjacent sequences:  A213778 A213779 A213780 * A213782 A213783 A213784

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jun 22 2012

STATUS

approved

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Last modified November 15 09:03 EST 2019. Contains 329144 sequences. (Running on oeis4.)