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A213548 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = m(m+1)/2, m = n-1+h, n>=1, h>=1, and ** = convolution. 6
1, 5, 3, 15, 12, 6, 35, 31, 22, 10, 70, 65, 53, 35, 15, 126, 120, 105, 81, 51, 21, 210, 203, 185, 155, 115, 70, 28, 330, 322, 301, 265, 215, 155, 92, 36, 495, 486, 462, 420, 360, 285, 201, 117, 45, 715, 705, 678, 630, 560, 470, 365, 253, 145, 55, 1001 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Principal diagonal: A213549.
Antidiagonal sums: A051836.
Row 1, (1,2,3,...)**(1,3,6,...): A000332.
Row 2, (1,2,3,...)**(3,6,10,...): A005718.
Row 3, (1,2,3,...)**(6,10,15,...): k*(k+1)*(k^2 + 13*k + 58)/24.
For a guide to related arrays, see A213500.
LINKS
Clark Kimberling, Antidiagonals 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) - 2*((n-1)^2)*x + n*(n-1)*x^2 and g(x) = 2*(1 - x)^5.
EXAMPLE
Northwest corner (the array is read by falling antidiagonals):
. 1, 5, 15, 35, 70, ...
. 3, 12, 31, 65, 120, ...
. 6, 22, 53, 105, 185, ...
. 10, 35, 81, 155, 265, ...
. 15, 51, 115, 215, 360, ...
. 21, 70, 155, 285, 470, ...
...
T(5,1) = (1)**(15) = 15;
T(5,2) = (1,2)**(15,21) = 1*21 + 2*15 = 51;
T(5,3) = (1,2,3)**(15,21,28) = 1*28 + 2*21 + 3*15 = 115;
T(4,4) = (1,2,3,4)**(10,15,21,28) = 1*28 + 2*21 + 3*15 + 4*10 = 155.
MATHEMATICA
b[n_] := n; c[n_] := n (n + 1)/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}] (* A213548 *)
d = Table[t[n, n], {n, 1, 40}] (* A213549 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A051836 *)
CROSSREFS
Cf. A213500.
Sequence in context: A146934 A213762 A178067 * A246204 A185733 A073677
KEYWORD
nonn,tabl,easy
AUTHOR
Clark Kimberling, Jun 16 2012
STATUS
approved

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Last modified July 21 18:17 EDT 2024. Contains 374475 sequences. (Running on oeis4.)