login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A323222 A(n, k) = [x^k] (1 - 4*x)^(-n/2)*x/(1 - x), square array read by ascending antidiagonals with n >= 0 and k >= 0. 5
0, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 5, 9, 1, 0, 1, 7, 21, 29, 1, 0, 1, 9, 37, 85, 99, 1, 0, 1, 11, 57, 177, 341, 351, 1, 0, 1, 13, 81, 313, 807, 1365, 1275, 1, 0, 1, 15, 109, 501, 1593, 3579, 5461, 4707, 1, 0, 1, 17, 141, 749, 2811, 7737, 15591, 21845, 17577, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,9
COMMENTS
General asymptotic formula for g.f. (1 - 4*x)^(-j/2)*x/(1 - x) and fixed j>0 is a(n) ~ n^(j/2 - 1) * 4^n / (3*Gamma(j/2)). - Vaclav Kotesovec, Jan 29 2019
LINKS
FORMULA
For n>0 and k>0 let X(n, k) denote the set of all tuples of length n with elements from {0, ..., k-1} with sum < k. Let b(m) = binomial(2*m, m). Then A(n, k) = Sum_{(j1,...,jn) in X(n, k)} b(j1)*b(j2)*...*b(jn).
EXAMPLE
[n\k] 0 1 2 3 4 5 6 7 8 9
-------------------------------------------------------------------
[0] 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A057427
[1] 0, 1, 3, 9, 29, 99, 351, 1275, 4707, 17577, ... A006134
[2] 0, 1, 5, 21, 85, 341, 1365, 5461, 21845, 87381, ... A002450
[3] 0, 1, 7, 37, 177, 807, 3579, 15591, 67071, 285861, ... A277178
[4] 0, 1, 9, 57, 313, 1593, 7737, 36409, 167481, 757305, ... A014916
[5] 0, 1, 11, 81, 501, 2811, 14823, 74883, 366603, 1752273, ... A323223
[6] 0, 1, 13, 109, 749, 4589, 26093, 140781, 730605, 3679725, ...
[7] 0, 1, 15, 141, 1065, 7071, 43107, 247311, 1355847, 7175661, ...
[8] 0, 1, 17, 177, 1457, 10417, 67761, 411825, 2377905, 13191345, ...
[9] 0, 1, 19, 217, 1933, 14803, 102319, 656587, 3982195, 23104441, ...
Triangle given by antidiagonals:
0;
0, 1;
0, 1, 1;
0, 1, 3, 1;
0, 1, 5, 9, 1;
0, 1, 7, 21, 29, 1;
0, 1, 9, 37, 85, 99, 1;
0, 1, 11, 57, 177, 341, 351, 1;
0, 1, 13, 81, 313, 807, 1365, 1275, 1;
0, 1, 15, 109, 501, 1593, 3579, 5461, 4707, 1;
MAPLE
Row := proc(n, len) local ogf, ser; ogf := (1 - 4*x)^(-n/2)*x/(1 - x);
ser := series(ogf, x, (n+1)*len+1); seq(coeff(ser, x, j), j=0..len) end:
for n from 0 to 9 do Row(n, 9) od;
MATHEMATICA
BF[N_, K_] := Module[{}, r[n_, k_] := FrobeniusSolve[ConstantArray[1, n], k];
X[n_] := Flatten[Table[r[N, j], {j, 0, n - 1}], 1];
CentralBinomial[n_] := Binomial[2 n, n];
Sum[Product[CentralBinomial[m[[i]]], {i, 1, N}], {m , X[K]}]];
Trow[n_] := Table[BF[n, k], {k, 0, 9}]; Table[Trow[n], {n, 1, 9}]
CROSSREFS
Sums of antidiagonals are A323217. Main diagonal is A323219.
Rows: A057427 (n=0), A006134 (n=1), A002450 (n=2), A277178 (n=3), A014916 (n=4), A323223 (n=5).
Columns: A005408 (k=2), A059993 (k=3), A323218 (k=4).
Similar array based on Catalan numbers is A323224.
Sequence in context: A121314 A119271 A364310 * A125104 A098157 A293617
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jan 24 2019
STATUS
approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 24 22:37 EDT 2024. Contains 374585 sequences. (Running on oeis4.)