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%I #41 Mar 21 2019 01:07:15
%S 1,2,0,2,9,3,18,3,9,64,48,200,96,200,48,64,625,750,2775,2280,4300,
%T 2280,2775,750,625,7776,12960,46440,53640,100584,81360,100584,53640,
%U 46440,12960,7776,117649,252105,909979,1337700,2594501,2753415,3604342,2753415,2594501,1337700,909979,252105,117649,2097152,5505024,20414464,36040704,73543680,94730496,133244544,128389632,133244544,94730496,73543680,36040704,20414464,5505024,2097152
%N Triangle, read by rows, where the g.f. of row n equals Product_{k=0..n-2} (n + k*y + n*y^2) for n > 1 with a single '1' in row 1.
%H Paul D. Hanna, <a href="/A324305/b324305.txt">Table of n, a(n) for n = 1..10000 terms in rows 1..100 of this triangle in flattened form.</a>
%F GENERATING FUNCTIONS.
%F E.g.f.: A(x,y) = x/(1 - y*A(x,y))^(1/y + y).
%F E.g.f.: A(x,y) = Series_Reversion( x*(1 - x*y)^(1/y + y) ), wrt x.
%F E.g.f.: A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2)
%F E.g.f.: A(x,y) = Series_Reversion( x/G(x,y) ) such that A(x/G(x,y),y) = x, where G(x,y) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1 + k*y + y^2) is the e.g.f. of A201949.
%F PARTICULAR ARGUMENTS.
%F E.g.f. at y = 0: A(x,y=0) = -LambertW(-x) = x*exp(-LambertW(-x)).
%F E.g.f. at y = 1: A(x,y=1) = x*G(x)^2, where G = 1 + x*G(x)^3 is the g.f. of A001764.
%F FORMULAS INVOLVING TERMS.
%F Row sums: Sum_{k=0..2*n-2} T(n,k) = (3*n-2)!/(2*n-1)! for n >= 1.
%F T(n,0) = T(n,2*n-2) = n^(n-1) for n >= 1.
%F T(n,n-1) = A324304(n) for n >= 1.
%e E.g.f.: A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2) and satisfies A(x,y) = x/(1 - y*A(x,y))^(1/y + y).
%e Explicitly,
%e A(x,y) = x + (2*y^2 + 2)*x^2/2! + (9*y^4 + 3*y^3 + 18*y^2 + 3*y + 9)*x^3/3! + (64*y^6 + 48*y^5 + 200*y^4 + 96*y^3 + 200*y^2 + 48*y + 64)*x^4/4! + (625*y^8 + 750*y^7 + 2775*y^6 + 2280*y^5 + 4300*y^4 + 2280*y^3 + 2775*y^2 + 750*y + 625)*x^5/5! + (7776*y^10 + 12960*y^9 + 46440*y^8 + 53640*y^7 + 100584*y^6 + 81360*y^5 + 100584*y^4 + 53640*y^3 + 46440*y^2 + 12960*y + 7776)*x^6/6! + (117649*y^12 + 252105*y^11 + 909979*y^10 + 1337700*y^9 + 2594501*y^8 + 2753415*y^7 + 3604342*y^6 + 2753415*y^5 + 2594501*y^4 + 1337700*y^3 + 909979*y^2 + 252105*y + 117649)*x^7/7! + ...
%e Setting y = 1 yields an o.g.f. of A006013:
%e A(x,y=1) = x + 2*x^2 + 7*x^3 + 30*x^4 + 143*x^5 + 728*x^6 + 3876*x^7 + 21318*x^8 + 120175*x^9 + ... + binomial(3*n-2,n-1)/n * x^n + ...
%e TRIANGLE.
%e This triangle of coefficients in Product_{k=0..n-2} (n + k*y + n*y^2), n >= 1, begins
%e 1;
%e 2, 0, 2;
%e 9, 3, 18, 3, 9;
%e 64, 48, 200, 96, 200, 48, 64;
%e 625, 750, 2775, 2280, 4300, 2280, 2775, 750, 625;
%e 7776, 12960, 46440, 53640, 100584, 81360, 100584, 53640, 46440, 12960, 7776;
%e 117649, 252105, 909979, 1337700, 2594501, 2753415, 3604342, 2753415, 2594501, 1337700, 909979, 252105, 117649;
%e 2097152, 5505024, 20414464, 36040704, 73543680, 94730496, 133244544, 128389632, 133244544, 94730496, 73543680, 36040704, 20414464, 5505024, 2097152; ...
%e RELATED SERIES.
%e The e.g.f. may be defined by A(x,y) = Series_Reversion( x/G(x,y) )
%e where G(x,y) is the e.g.f. of A201949 and equals
%e G(x,y) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1 + k*y + y^2)
%e so that
%e G(x,y) = 1 + (1 + y^2)*x + (1 + y + 2*y^2 + y^3 + y^4)*x^2/2! + (1 + 3*y + 5*y^2 + 6*y^3 + 5*y^4 + 3*y^5 + y^6)*x^3/3! + (1 + 6*y + 15*y^2 + 24*y^3 + 28*y^4 + 24*y^5 + 15*y^6 + 6*y^7 + y^8)*x^4/4! + (1 + 10*y + 40*y^2 + 90*y^3 + 139*y^4 + 160*y^5 + 139*y^6 + 90*y^7 + 40*y^8 + 10*y^9 + y^10)*x^5/5! + ...
%e and G(x,y) = x / Series_Reversion( A(x,y) ).
%e RELATED TRIANGLE.
%e Triangle A201949 of coefficients in G(x,y) such that A(x/G(x,y),y) = x begins
%e 1;
%e 1, 0, 1;
%e 1, 1, 2, 1, 1;
%e 1, 3, 5, 6, 5, 3, 1;
%e 1, 6, 15, 24, 28, 24, 15, 6, 1;
%e 1, 10, 40, 90, 139, 160, 139, 90, 40, 10, 1;
%e 1, 15, 91, 300, 629, 945, 1078, 945, 629, 300, 91, 15, 1; ...
%e where the g.f. of row n is Product_{k=0..n-1} (1 + k*y + y^2) for n >= 0.
%o (PARI) {T(n, k)=polcoeff(prod(j=0, n-2, n + j*y + n*y^2), k, y)}
%o {for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))}
%o (PARI) /* A(x,y) = Series_Reversion(x/G(x,y)) where G(x,y) = e.g.f. A201949 */
%o {T(n,k) = my(G=1,A=x);
%o G = sum(m=0,n, x^m/m! * prod(j=0,m-1, 1 + j*y + y^2) +x*O(x^n));
%o A = serreverse(x/G);
%o n!*polcoeff(polcoeff(A,n,x),k,y)}
%o {for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))}
%Y Cf. A201949, A324304.
%Y Cf. A324956, A324958.
%K nonn,tabf
%O 1,2
%A _Paul D. Hanna_, Feb 28 2019
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