login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A213558 Rectangular array:  (row n) = b**c, where b(h) = h^3, c(h) = (n-1+h)^3, n>=1, h>=1, and ** = convolution. 4
1, 16, 8, 118, 91, 27, 560, 496, 280, 64, 2003, 1878, 1366, 637, 125, 5888, 5672, 4672, 2944, 1216, 216, 14988, 14645, 12917, 9542, 5446, 2071, 343, 34176, 33664, 30920, 25088, 17088, 9088, 3256, 512, 71445, 70716, 66620, 57359, 43535 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Principal diagonal:  A213559

Antidiagonal sums:  A213560

Row 1,  (1,8,27,...)**(1,8,27,...):  A145216

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) = 8*T(n,k-1) - 28*T(n,k-2) + 56*T(n,k-3) - 70*T(n,k-4) + 56*T(n,k-5) - 28*T(n,k-6) + 8*T(n,k-7) - T(n,k-8).

G.f. for row n:  f(x)/g(x), where f(x) = n^3 + ((n + 1)^3)*x + (-8*n^3 + 6*n^2 + 12*n + 8)*x^2 + (8*n^3 - 18*n^2 + 18)*x^3 - ((n - 2)^3)*x^4 - ((n + 1)^3)*x^5 and g(x) = (1 - x)^8.

EXAMPLE

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

1.....16.....118....560.....2003

8.....91.....496....1878....5672

27....280....1366...4672....12917

64....637....2944...9542....25088

125...1216...5446...17088...43535

MATHEMATICA

b[n_] := n^3; c[n_] := n^3

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}]  (* A213558 *)

d = Table[t[n, n], {n, 1, 40}] (* A213559 *)

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

s1 = Table[s[n], {n, 1, 50}] (* A213560 *)

CROSSREFS

Cf. A213500.

Sequence in context: A258160 A040243 A299584 * A164703 A153724 A183396

Adjacent sequences:  A213555 A213556 A213557 * A213559 A213560 A213561

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jun 17 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 12 19:25 EST 2019. Contains 329078 sequences. (Running on oeis4.)