A030523 A convolution triangle of numbers obtained from A001792.
1, 3, 1, 8, 6, 1, 20, 25, 9, 1, 48, 88, 51, 12, 1, 112, 280, 231, 86, 15, 1, 256, 832, 912, 476, 130, 18, 1, 576, 2352, 3276, 2241, 850, 183, 21, 1, 1280, 6400, 10976, 9424, 4645, 1380, 245, 24, 1, 2816, 16896, 34848, 36432, 22363, 8583, 2093, 316, 27, 1
Offset: 1
Examples
{1}; {3,1}; {8,6,1}; {20,25,9,1}; {48,88,51,12,1}; ...
(0, 3, -1/3, 4/3, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
1
0 1
0 3 1
0 8 6 1
0 20 25 9 1
0 48 88 51 12 1
...
- _Philippe Deléham_, Feb 20 2013
Links
- W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
- W. Lang, First ten rows.
Crossrefs
Cf. A057682 (alternating row sums).
Programs
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Mathematica
a[n_, m_] := SeriesCoefficient[(1-2*x)^2/((x^2-x)*y + (1-2*x)^2) - 1, {x, 0, n}, {y, 0, m}]; Table[a[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Apr 28 2015, after Vladimir Kruchinin *)
Formula
a(n, 1) = A001792(n-1).
Row sums = A039717(n).
a(n, m) = 2*(2*m+n-1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n
T(n,k) = 4*T(n-1,k) - 4*T(n-2,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = 0, T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Feb 20 2013
Sum_{k=1..n} T(n,k)*2^(k-1) = A140766(n). -Philippe Deléham, Feb 20 2013
G.f.: (1-2*x)^2/((x^2-x)*y+(1-2*x)^2)-1. - Vladimir Kruchinin, Apr 28 2015
A161727 a(n) = ((2+sqrt(3))*(4+sqrt(3))^n-(2-sqrt(3))*(4-sqrt(3))^n)/sqrt(12).
1, 6, 35, 202, 1161, 6662, 38203, 219018, 1255505, 7196806, 41252883, 236464586, 1355429209, 7769394054, 44534572715, 255274459018, 1463246226849, 8387401847558, 48077013831427, 275579886633162, 1579637913256745
Offset: 0
Links
- Harvey P. Dale, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (8,-13).
Programs
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Maple
seq(expand(((2+sqrt(3))*(4+sqrt(3))^n-(2-sqrt(3))*(4-sqrt(3))^n)/sqrt(12)), n = 0 .. 20) # Emeric Deutsch, Jun 20 2009
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Mathematica
LinearRecurrence[{8,-13},{1,6},30] (* Harvey P. Dale, Jun 01 2016 *) -
PARI
F=nfinit(x^2-3); for(n=0, 20, print1(nfeltdiv(F, ((2+x)*(4+x)^n-(2-x)*(4-x)^n), (2*x))[1], ",")) \\ Klaus Brockhaus, Jun 19 2009
Formula
a(n) = 8*a(n-1)-13(n-2) for n > 1; a(0) = 1, a(1) = 6.
G.f.: (1-2*x)/(1-8*x+13*x^2). - Klaus Brockhaus, Jun 19 2009
Extensions
Extended beyond a(6) by Klaus Brockhaus and Emeric Deutsch, Jun 19 2009
Edited by Klaus Brockhaus, Jul 05 2009
Comments