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1, 3, 18, 160, 1752, 21504, 282304, 3867840, 54547200, 785255680, 11478167040, 169748686848, 2533556365312, 38094656593920, 576271774875648, 8761529890717696, 133776598692003840, 2050020136793604096
FORMULA
G.f.: ge(x^2) + x*go(x^2) with ge(x) g.f. of A067303 and go(x) g.f. of A067322.
Triangle of coefficients of polynomials used for g.f.s of columns of A067304.
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1, 3, 1, 3, 10, 3, 3, 10, 12, 3, 21, 70, 93, 60, 12, 48, 160, 219, 165, 72, 12, 507, 1690, 2343, 1872, 996, 336, 48, 1461, 4870, 6798, 5595, 3252, 1380, 384, 48, 17841, 59470, 83361, 69828, 42636, 20256, 7248, 1728
COMMENTS
The row polynomials p(n,y) := sum(a(n,m)y^m,m=0..n), n>=1, appear in the g.f.s for the n-th column of triangle A067304.
FORMULA
a(n, m)=[y^m](p(n, y)), n>=m>=1, a(0, 0)=1, else 0, where p(k, y) is, for k>=1, defined by the g.f. of the k-th column of triangle A067304(n, k).
EXAMPLE
{1}; {3,1}; {3,10,3}; {3,3,10,12}; ...
1, 5, 36, 328, 3440, 39408, 478912, 6068480, 79315200, 1061628160, 14480086016, 200540018688, 2812618092544, 39867889037312, 570237130752000, 8219880968028160, 119293333282291712, 1741605394647416832
FORMULA
G.f.: 4*(3+c(4*x))*(c(4*x)^3)/(1+c(4*x))^4 with c(x) g.f. of A000108 (Catalan).
One-fourth of third column of triangle A067304.
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0
1, 8, 75, 796, 9176, 111936, 1421968, 18618560, 249542400, 3407171584, 47226230528, 662805371904, 9400304896000, 134517761982464, 1939837469085696, 28162286932246528, 411276783645753344
FORMULA
G.f.: (3+10c(4*x)+3*c(4*x)^2)*(c(4*x)^3)/(1+c(4*x))^4, with c(x) g.f. of A000108 (Catalan).
One-fourth of fourth column of triangle A067304.
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7, 71, 768, 8920, 109232, 1390800, 18237952, 244701440, 3343713024, 46374830848, 651170275328, 9238908291072, 132251092529152, 1907671386263552, 27701755840561152, 404632598092447744
FORMULA
G.f.: 4(3+10*c(4*x)+12*c(4*x)^2+3*c(4*x)^3)*(c(4*x)^3)/(1+c(4*x))^4, with c(x) g.f. of A000108 (Catalan).
One sixteenth of fifth column of triangle A067304.
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16, 185, 2181, 26860, 342968, 4504944, 60509296, 827456576, 11482655232, 161302619392, 2289365653760, 32780329073664, 472951175022592, 6869148315201536, 100352220112662528, 1473672361011920896
FORMULA
G.f.: (21+70c(4*x)+93*c(4*x)^2+60*c(4*x)^3+12*c(4*x)^4)*(c(4*x)^3)/(1+c(4*x))^4, with c(x) g.f. of A000108 (Catalan).
EXAMPLE
21+70*y+93*y^2+60*y^3 = p(4,y), fifth row polynomial of triangle A067329.
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