OFFSET
0,5
COMMENTS
FORMULA
The g.f. of the row sums, A(x,1), equals the limit of nested squares given by
A(x,1) = 1 + x*(1 + 2*x*(1 + 3*x*(1 + 4*x*(...(1 + n*x*(...)^2)^2...)^2)^2)^2)^2.
EXAMPLE
This triangle of coefficients in g.f. A(x,y) begins:
1;
0, 1;
0, 2, 2;
0, 9, 14, 5;
0, 64, 124, 74, 14;
0, 624, 1388, 1074, 352, 42;
0, 7736, 18964, 17292, 7520, 1588, 132;
0, 116416, 307088, 314356, 163728, 46561, 6946, 429;
0, 2060808, 5760704, 6434394, 3807910, 1311172, 266116, 29786, 1430;
0, 41952600, 122980872, 147159406, 95921164, 37846790, 9373620, 1438006, 126008, 4862;
0, 965497440, 2945806672, 3729264888, 2623904244, 1147995184, 327833296, 61731036, 7455440, 527900, 16796;
0, 24786054816, 78270032288, 103887986400, 77816220888, 36954748286, 11761455804, 2565654006, 382043344, 37445610, 2195580, 58786; ...
where the g.f. A(x,y) = 1 + x*y*A(x,y+1)^2 begins:
A(x,y) = 1 + x*(y) + x^2*(2*y + 2*y^2) +
x^3*(9*y + 14*y^2 + 5*y^3) +
x^4*(64*y + 124*y^2 + 74*y^3 + 14*y^4) +
x^5*(624*y + 1388*y^2 + 1074*y^3 + 352*y^4 + 42*y^5) +
x^6*(7736*y + 18964*y^2 + 17292*y^3 + 7520*y^4 + 1588*y^5 + 132*y^6) +
x^7*(116416*y + 307088*y^2 + 314356*y^3 + 163728*y^4 + 46561*y^5 + 6946*y^6 + 429*y^7) +
x^8*(2060808*y + 5760704*y^2 + 6434394*y^3 + 3807910*y^4 + 1311172*y^5 + 266116*y^6 + 29786*y^7 + 1430*y^8) +...
RELATED TRIANGLES.
The triangle T1 of coefficients in A(x,y+1) begins:
1;
1, 1;
4, 6, 2;
28, 52, 29, 5;
276, 590, 430, 130, 14;
3480, 8240, 7142, 2902, 562, 42;
53232, 136352, 133820, 65892, 17440, 2380, 132;
955524, 2606056, 2811333, 1588813, 515738, 97246, 9949, 429;
19672320, 56489536, 65680352, 41222664, 15498120, 3613454, 514658, 41226, 1430;
456803328, 1369670752, 1692959656, 1154579428, 485522796, 131955696, 23376294, 2621102, 169766, 4862;
11810032896, 36744177952, 47799342376, 34885949644, 16033889224, 4899599348, 1016573628, 142394476, 12962360, 695860, 16796; ...
in which row sums form A128571:
[1, 2, 12, 114, 1440, 22368, 409248, 8585088, ...]
where
A(x,y+1) = 1 + x*(1 + y) + x^2*(4 + 6*y + 2*y^2) +
x^3*(28 + 52*y + 29*y^2 + 5*y^3) +
x^4*(276 + 590*y + 430*y^2 + 130*y^3 + 14*y^4) +
x^5*(3480 + 8240*y + 7142*y^2 + 2902*y^3 + 562*y^4 + 42*y^5) +
x^6*(53232 + 136352*y + 133820*y^2 + 65892*y^3 + 17440*y^4 + 2380*y^5 + 132*y^6) +
x^7*(955524 + 2606056*y + 2811333*y^2 + 1588813*y^3 + 515738*y^4 + 97246*y^5 + 9949*y^6 + 429*y^7) +...
The triangle T2 of coefficients in A(x,y)^2 begins:
1;
0, 2;
0, 4, 5;
0, 18, 32, 14;
0, 128, 270, 184, 42;
0, 1248, 2940, 2488, 928, 132;
0, 15472, 39513, 38364, 18266, 4372, 429;
0, 232832, 633296, 678712, 377332, 117430, 19776, 1430;
0, 4121616, 11800512, 13648092, 8478840, 3119480, 692086, 87112, 4862;
0, 83905200, 250768144, 308424612, 208690548, 86565216, 22913292, 3836896, 376736, 16796;
0, 1930994880, 5987236848, 7750642944, 5617656996, 2555316840, 767744018, 154465024, 20330760, 1607720, 58786; ...
in which row sums form A128577:
[1, 2, 9, 64, 624, 7736, 116416, 2060808, 41952600, ...]
where
A(x,y)^2 = 1 + x*(2*y) + x^2*(4*y + 5*y^2) +
x^3*(18*y + 32*y^2 + 14*y^3) +
x^4*(128*y + 270*y^2 + 184*y^3 + 42*y^4) +
x^5*(1248*y + 2940*y^2 + 2488*y^3 + 928*y^4 + 132*y^5) +
x^6*(15472*y + 39513*y^2 + 38364*y^3 + 18266*y^4 + 4372*y^5 + 429*y^6) +
x^7*(232832*y + 633296*y^2 + 678712*y^3 + 377332*y^4 + 117430*y^5 + 19776*y^6 + 1430*y^7) +...
PROG
(PARI) /* Print this triangle of coefficients in A(x, y): */
{T(n, k) = my(A=1); for(i=1, n, A = 1 + x*y*subst(A, y, y+1)^2 +x*O(x^n)); polcoeff(polcoeff(A, n, x), k, y)}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) /* Print triangle of coefficients in A(x, y+1): */
{T1(n, k) = my(A=1); for(i=1, n, A = 1 + x*y*subst(A, y, y+1)^2 +x*O(x^n)); polcoeff(polcoeff(subst(A, y, y+1), n, x), k, y)}
for(n=0, 12, for(k=0, n, print1(T1(n, k), ", ")); print(""))
(PARI) /* Print triangle of coefficients in A(x, y)^2: */
{T2(n, k) = my(A=1); for(i=1, n, A = 1 + x*y*subst(A, y, y+1)^2 +x*O(x^n)); polcoeff(polcoeff(A^2, n, x), k, y)}
for(n=0, 12, for(k=0, n, print1(T2(n, k), ", ")); print(""))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Mar 16 2016
STATUS
approved