OFFSET
1,6
FORMULA
G.f. A(x,2) = C(x) = x + C(x^2 + 2*x*C(x)^2) where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
EXAMPLE
Given A(x,y) = x + A( x^2 + x*y*A(x,y)^2, y), then the coefficient of x^n in A(x,y) begins:
n=1: 1;
n=2: 1;
n=3: y;
n=4: 1 + 2*y;
n=5: 3*y + 2*y^2;
n=6: 7*y + 7*y^2;
n=7: 4*y + 21*y^2 + 5*y^3;
n=8: 1 + 6*y + 46*y^2 + 29*y^3;
n=9: 9*y + 65*y^2 + 114*y^3 + 15*y^4;
n=10: 13*y + 113*y^2 + 304*y^3 + 122*y^4;
n=11: 8*y + 169*y^2 + 649*y^3 + 582*y^4 + 50*y^5;
n=12: 19*y + 229*y^2 + 1311*y^3 + 1931*y^4 + 514*y^5;
n=13: 14*y + 326*y^2 + 2289*y^3 + 5235*y^4 + 2915*y^5 + 177*y^6;
n=14: 4*y + 511*y^2 + 3800*y^3 + 12353*y^4 + 11667*y^5 + 2179*y^6;
n=15: 8*y + 528*y^2 + 6365*y^3 + 25663*y^4 + 37605*y^5 + 14439*y^6 + 651*y^7;
n=16: 1 + 14*y + 602*y^2 + 9933*y^3 + 50117*y^4 + 102960*y^5 + 67567*y^6 + 9313*y^7;
n=17: 17*y + 779*y^2 + 13887*y^3 + 93176*y^4 + 249123*y^5 + 251277*y^6 + 70851*y^7 + 2461*y^8;
n=18: 27*y + 822*y^2 + 19953*y^3 + 161702*y^4 + 554778*y^5 + 787255*y^6 + 378828*y^7 + 40107*y^8;
n=19: 20*y + 985*y^2 + 26748*y^3 + 267548*y^4 + 1149904*y^5 + 2169902*y^6 + 1596301*y^7 + 344833*y^8 + 9503*y^9;
n=20: 33*y + 1423*y^2 + 33547*y^3 + 428642*y^4 + 2237223*y^5 + 5425404*y^6 + 5639060*y^7 + 2072343*y^8 + 173817*y^9; ...
where the coefficients of x^n at y=2 yield the Catalan sequence (A000108)
and the coefficients of x^n at y=1 yield sequence A271867.
This table begins:
1: [1],
2: [1],
3: [0, 1],
4: [1, 2],
5: [0, 3, 2],
6: [0, 7, 7],
7: [0, 4, 21, 5],
8: [1, 6, 46, 29],
9: [0, 9, 65, 114, 15],
10: [0, 13, 113, 304, 122],
11: [0, 8, 169, 649, 582, 50],
12: [0, 19, 229, 1311, 1931, 514],
13: [0, 14, 326, 2289, 5235, 2915, 177],
14: [0, 4, 511, 3800, 12353, 11667, 2179],
15: [0, 8, 528, 6365, 25663, 37605, 14439, 651],
16: [1, 14, 602, 9933, 50117, 102960, 67567, 9313],
17: [0, 17, 779, 13887, 93176, 249123, 251277, 70851, 2461],
18: [0, 27, 822, 19953, 161702, 554778, 787255, 378828, 40107],
19: [0, 20, 985, 26748, 267548, 1149904, 2169902, 1596301, 344833, 9503],
20: [0, 33, 1423, 33547, 428642, 2237223, 5425404, 5639060, 2072343, 173817],
21: [0, 22, 1696, 45001, 644977, 4148095, 12510282, 17417722, 9761246, 1666931, 37325],
22: [0, 8, 1951, 60518, 941911, 7327901, 27001551, 48380186, 38383316, 11121058, 757166],
23: [0, 16, 2032, 76469, 1368689, 12325683, 55128925, 123212108, 131265572, 57914532, 8013226, 148658],
24: [0, 43, 2233, 97715, 1929992, 20063142, 106847213, 292161779, 401413381, 250837500, 58766538, 3312223],
25: [0, 26, 2676, 122275, 2671266, 31693646, 197758824, 651604747, 1120119759, 940861815, 335281883, 38344863, 598978],
26: [0, 14, 3186, 146875, 3690225, 48570293, 352082741, 1376271666, 2895874917, 3142495637, 1585770660, 306614741, 14540518],
27: [0, 28, 3332, 177768, 4955379, 73062941, 604474079, 2771307598, 7012061147, 9538931973, 6473312499, 1901935380, 182787816, 2437164],
28: [0, 4, 4347, 218560, 6494263, 107933186, 1005837512, 5347520176, 16035090718, 26718656916, 23451518737, 9753064433, 1582783800, 64024175],
29: [0, 0, 4526, 267616, 8499808, 155611109, 1631859790, 9927108488, 34865937677, 69852304346, 76953296350, 43045676583, 10605749919, 868524649, 9999912],
30: [0, 8, 3228, 330909, 11026402, 220574099, 2583126916, 17816436623, 72459675808, 171977635171, 232245199109, 168136944474, 58609642777, 8096622083, 282639031],
31: [0, 16, 3680, 378032, 14174509, 308537087, 3995815653, 31032068182, 144593007264, 401550149642, 652283763087, 593181170120, 278108736935, 58282610197, 4115397063, 41329076],
32: [1, 30, 4274, 422797, 18166709, 424609949, 6063752715, 52580292690, 278252968814, 894172441003, 1720698824242, 1919504552856, 1165137737724, 345292508754, 41094699891, 1250545089], ...
PROG
(PARI) {T(n, k) = my(A=x+x^2 +x*O(x^n)); for(i=1, n, A = x + subst(A, x, x^2 + y*x*A^2 +x*O(x^n)) ) ; polcoeff(polcoeff(A, n, x), k, y)}
for(n=1, 32, for(k=0, (n-1)\2, print1(T(n, k), ", ")); print(""))
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Paul D. Hanna, Apr 16 2016
STATUS
approved