Original entry on oeis.org
0, 0, 0, 10, 2574, 748616, 282846568, 141482705378, 93129791442534, 79703248816409088, 87547852413089111888, 121899005847224540133690, 212656111085108159911203710, 459737640779469178164281972792, 1218867518038879445116138285206008, 3923403293626677106527196012373423122
Offset: 0
-
CL := (f, x) -> PolynomialTools:-CoefficientList(f, x):
A327002 := proc(n) local P, Q;
P := proc(n) option remember; if n = 0 then return 1 fi;
add(binomial(3*n, 3*k+3)*P(n-k-1)*x, k=0..n-1) end:
Q := proc(n) option remember; if n = 0 then return 1 fi;
add(binomial(3*n-1, 3*k)*Q(k)*Q(n-1-k), k=0..n-1) end:
Q(n) - add(CL(P(n),x)[k+1]/k!, k=0..n) end:
seq(A327002(n), n=0..15);
A327000
A(n, k) = A309522(n, k) - A327001(n, k) for n >= 0 and k >= 3, square array read by ascending antidiagonals.
Original entry on oeis.org
1, 1, 6, 3, 9, 26, 10, 117, 68, 100, 35, 2574, 4500, 517, 365, 126, 70005, 748616, 199155, 4163, 1302, 462, 2082759, 192426260, 282846568, 10499643, 36180, 4606, 1716, 65061234, 59688349943, 799156187475, 141482705378, 663488532, 341733, 16284
Offset: 0
Array starts:
n\k [ 3 4 5 6 7 ]
[0] 1, 6, 26, 100, 365, ... [A125107]
[1] 1, 9, 68, 517, 4163, ... [A048742]
[2] 3, 117, 4500, 199155, 10499643, ... [A326995]
[3] 10, 2574, 748616, 282846568, 141482705378, ... [A327002]
[4] 35, 70005, 192426260, 799156187475, 4961959681629275, ...
[5] 126, 2082759, 59688349943, 3097220486457142, 278271624962638244163, ...
A001700,
-
ListTools:-Flatten([seq(seq(A309522(n-k, k) - A327001(n-k, k), k=3..n), n=3..10)]);
A309522
Generalized Blasius numbers, square array read by ascending antidiagonals, A(n, k) for n, k >= 0.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 4, 6, 14, 1, 1, 11, 34, 24, 42, 1, 1, 36, 375, 496, 120, 132, 1, 1, 127, 6306, 27897, 11056, 720, 429, 1, 1, 463, 129256, 3156336, 3817137, 349504, 5040, 1430, 1, 1, 1717, 2877883, 514334274, 3501788976, 865874115, 14873104, 40320, 4862
Offset: 0
Table A(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
[0] 1, 1, 2, 5, 14, 42, 132, ... A000108
[1] 1, 1, 2, 6, 24, 120, 720, ... A000142
[2] 1, 1, 4, 34, 496, 11056, 349504, ... A002105
[3] 1, 1, 11, 375, 27897, 3817137, 865874115, ... A018893
[4] 1, 1, 36, 6306, 3156336, 3501788976, 7425169747776, ...
A260878|
- Heinrich Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. u. Physik 56 (1908), 1-37; see p. 8. [This article was based on his PhD thesis. He corrected c_6 = A(n=3, k=6) but his "correction" of c_7 = A(n=3, k=7) was not right!]
- Heinrich Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. u. Physik 56 (1908), 1-37 [English translation by J. Vanier on behalf of the National Advisory Committee for Aeronautics (NACA), 1950]; see p. 8. [This is a translation of Blasius' article. The value of c_6 = A(n=3, k=6) was corrected in the article and the translation, but the "correction" for c_7 = A(n=3, k=7) in both documents is wrong.]
- Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014.
- Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, Discrete Mathematics 339(1) (2016), 227-254.
- Hans Salié, Über die Koeffizienten der Blasiusschen Reihen, Math. Nachr. 14 (1955), 241-248 (1956). [In the article the array is denoted by c^{(n)}_v for n, v >= 1. We have A(n, k) = c^{(n)}_{k+1} for n >= 1 and k >= 0. The Catalan numbers (row n = 0 for A(n, k)) do not appear in Salié's article.]
-
A := proc(n, k) option remember; if k = 0 then 1 else
add(binomial(n*k-1, n*v)*A(n, v)*A(n, k-1-v), v=0..k-1) fi end:
seq(seq(A(n-k, k), k=0..n), n=0..9);
-
A[n_, k_] := A[n, k] = If[k == 0, 1, Sum[Binomial[n*k - 1, n*v]*A[n, v]* A[n, k - 1 - v], {v, 0, k - 1}]];
Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 26 2019, from Maple *)
Showing 1-3 of 3 results.
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