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A213505 Rectangular array:  (row n) = b**c, where b(h) = h^2, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution. 4
1, 8, 4, 34, 25, 9, 104, 88, 52, 16, 259, 234, 170, 89, 25, 560, 524, 424, 280, 136, 36, 1092, 1043, 899, 674, 418, 193, 49, 1968, 1904, 1708, 1384, 984, 584, 260, 64, 3333, 3252, 2996, 2555, 1979, 1354, 778, 337, 81, 5368, 5268, 4944, 4368, 3584 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Principal diagonal: A213546.

Antidiagonal sums: A213547.

Row 1, (1,4,9,...)**(1,4,9,...): A033455.

Row 2, (1,4,9,...)**(4,9,16,...): (k^5 + 10*k^4 + 40*k^3 + 50*k^2 +19*k)/30.

Row 3, (1,4,9,...)**(9,16,25,...):  (k^5 + 15*k^4 + 90*k^3 + 120*k^2+44*k)/30.

For a guide to related arrays, see A213500.

LINKS

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

Henri Muehle, Proper Mergings of Stars and Chains are Counted by Sums of Antidiagonals in Certain Convolution Arrays -- The Details, arXiv preprint arXiv:1301.1654, 2013.

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 - (n^2 - 2*n - 1)*x - (n^2 - 2)*x^2 - ((n - 1)^2)*x^3 and g(x) = (1 - x)^6.

EXAMPLE

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

1....8.....34....104...259....560

4....25....88....234...524....1043

9....52....170...424...899....1708

16...89....280...674...1384...2555

25...136...418...984...1979...3584

...

T(5,1) = (1)**(25) = 25

T(5,2) = (1,4)**(25,36) = 1*36+4*25 = 136

T(5,3) = (1,4,9)**(25,36,49) = 1*49+4*36+9*25 = 418

MATHEMATICA

b[n_] := n^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}]  (* A213505 *)

d = Table[t[n, n], {n, 1, 40}] (* A213546 *)

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

s1 = Table[s[n], {n, 1, 50}] (* A213547 *)

CROSSREFS

Cf. A213500.

Sequence in context: A213178 A082682 A279635 * A270232 A270716 A270457

Adjacent sequences:  A213502 A213503 A213504 * A213506 A213507 A213508

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jun 16 2012

STATUS

approved

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Last modified April 26 08:16 EDT 2019. Contains 322472 sequences. (Running on oeis4.)