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A277410
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G.f. A(x,y) satisfies: A( x - y*G(x,y), y) = x + (1-y)*G(x,y) such that G(x,y) = Integral A(x,y) dx, where the coefficients T(n,k) of x^n*y^k form a triangle read by rows n>=1, for k=0..n-1.
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15
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1, 1, 0, 1, 3, 0, 1, 13, 15, 0, 1, 38, 165, 105, 0, 1, 94, 1033, 2310, 945, 0, 1, 213, 4953, 26229, 36330, 10395, 0, 1, 459, 20370, 213511, 674520, 640710, 135135, 0, 1, 960, 76056, 1421225, 8559675, 18127935, 12588345, 2027025, 0, 1, 1972, 266334, 8283234, 85654979, 337805535, 515903850, 273544425, 34459425, 0, 1, 4007, 892542, 44013478, 729292193, 4822487682, 13506364410, 15631793100, 6529047525, 654729075, 0
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OFFSET
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1,5
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COMMENTS
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More generally, we have the following related identity.
Given functions F and G with F(0)=0, F'(0)=1, G(0)=0, G'(0)=0,
if F(x - y*G(x)) = x + (1-y)*G(x), then
(1) F(x) = x + G( y*F(x) + (1-y)*x ),
(2) y*F(x) + (1-y)*x = Series_Reversion(x - y*G(x)),
(3) F(x) = x + G(x + y*G(x + y*G(x + y*G(x +...)))),
(4) F(x) = x + Sum_{n>=1} y^(n-1) * d^(n-1)/dx^(n-1) G(x)^n / n!.
The g.f. of this sequence A(x,y) equals F(x) in the above when G(x) = Integral F(x) dx.
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LINKS
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FORMULA
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Given g.f. A(x,y), define G(x,y) = Integral A(x,y) dx, then
(1) A(x,y) = x + G( y*A(x,y) + (1-y)*x, y),
(2) y*A(x,y) + (1-y)*x = Series_Reversion( x - y*G(x,y) ),
(3) y*x + (1-y)*B(x,y) = Series_Reversion( x + (1-y)*G(x,y) ), where B( A(x,y), y) = x.
(4) A(x,y) = x + Sum_{n>=1} y^(n-1) * d^(n-1)/dx^(n-1) G(x,y)^n / n!.
In formulas 2 and 3, the series reversion is taken with respect to variable x.
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EXAMPLE
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G.f.: A(x,y) = x + x^2/2! + (3*y + 1)*x^3/3! + (15*y^2 + 13*y + 1)*x^4/4! + (105*y^3 + 165*y^2 + 38*y + 1)*x^5/5! + (945*y^4 + 2310*y^3 + 1033*y^2 + 94*y+ 1)*x^6/6! + (10395*y^5 + 36330*y^4 + 26229*y^3 + 4953*y^2 + 213*y + 1)*x^7/7! + (135135*y^6 + 640710*y^5 + 674520*y^4 + 213511*y^3 + 20370*y^2 + 459*y + 1)*x^8/8! + (2027025*y^7 + 12588345*y^6 + 18127935*y^5 + 8559675*y^4 + 1421225*y^3 + 76056*y^2 + 960*y + 1)*x^9/9! + (34459425*y^8 + 273544425*y^7 + 515903850*y^6 + 337805535*y^5 + 85654979*y^4 + 8283234*y^3 + 266334*y^2 + 1972*y + 1)*x^10/10! +...
such that A( x - y*G(x,y), y) = x + (1-y)*G(x,y)
also,
A(x,y) = x + G( y*A(x,y) + (1-y)*x, y)
where G(x,y) = Integral A(x,y).
...
This triangle of coefficients T(n,k) of x^n*y^k/n! in g.f. A(x,y) begins:
1;
1, 0;
1, 3, 0;
1, 13, 15, 0;
1, 38, 165, 105, 0;
1, 94, 1033, 2310, 945, 0;
1, 213, 4953, 26229, 36330, 10395, 0;
1, 459, 20370, 213511, 674520, 640710, 135135, 0;
1, 960, 76056, 1421225, 8559675, 18127935, 12588345, 2027025, 0;
1, 1972, 266334, 8283234, 85654979, 337805535, 515903850, 273544425, 34459425, 0;
1, 4007, 892542, 44013478, 729292193, 4822487682, 13506364410, 15631793100, 6529047525, 654729075, 0;
1, 8089, 2900353, 218797958, 5531376285, 57226590953, 264482764305, 555756298020, 505173143475, 170116046100, 13749310575, 0; ...
in which the diagonal equals A001147 (odd double factorials), and the row sums yield A210949.
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APPLICATION.
Given F(x) such that
F(x - Integral p*F(x) dx) = x + Integral q*F(x) dx
then
F(x) = Sum_{n>=1} a(n)*x^n/n!
where
a(n) = Sum_{k=0..n-1} A277410(n,k) * p^k * (p+q)^(n-k-1) for n>=1.
EXAMPLES.
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COLUMN GENERATING FUNCTIONS.
Colin Barker observed that column 1 of this triangle (A277411) appears to have the o.g.f. x*(3*x-2*x^2) / ((1-x)^3*(1-2*x)).
This observation led to the following conjecture.
Let F(k,x) = o.g.f. of column k in this triangle,
then
F(k,x) = P(k,x) * x^(k+1) / Product_{j=0..k} (1 - (j+1)*x)^(2*(k-j)+1)
where P(k,x) is a polynomial in x with degree k*(k+1) for k>=0.
Example:
F(0,x) = x/(1-x) ;
F(1,x) = P(1,x)*x^2/((1-x)^3*(1-2*x)) ;
F(2,x) = P(2,x)*x^3/((1-x)^5*(1-2*x)^3*(1-3*x)) ;
F(3,x) = P(3,x)*x^4/((1-x)^7*(1-2*x)^5*(1-3*x)^3*(1-4*x)) ;
...
The polynomials P(k,x) begin:
P(0,x) = 1 ;
P(1,x) = 3*x - 2*x^2 ;
P(2,x) = 15*x - 45*x^2 - 2*x^3 + 106*x^4 - 92*x^5 + 24*x^6 ;
P(3,x) = 105*x - 840*x^2 + 504*x^3 + 16321*x^4 - 75880*x^5 + 154483*x^6 - 152077*x^7 + 39208*x^8 + 59000*x^9 - 60336*x^10 + 23328*x^11 - 3456*x^12 ;
P(4,x) = 945*x - 15645*x^2 + 32445*x^3 + 1255770*x^4 - 15120061*x^5 + 86803308*x^6 - 291640845*x^7 + 529758178*x^8 - 50236668*x^9 - 2553002523*x^10 + 7695202852*x^11 - 12713196156*x^12 + 13351222596*x^13 - 8752472980*x^14 + 2871967920*x^15 + 387984096*x^16 - 884504448*x^17 + 427064832*x^18 - 100694016*x^19 + 9953280*x^20 ;
P(5,x) = 10395*x - 305235*x^2 + 1299375*x^3 + 77300220*x^4 - 1834009998*x^5 + 21447595316*x^6 - 156933684108*x^7 + 721294719700*x^8 - 1490891586137*x^9 - 5868653004882*x^10 + 70213320019895*x^11 - 359261247450016*x^12 + 1234731543184308*x^13 - 3081038591203028*x^14 + 5553265322783926*x^15 - 6518085613542516*x^16 + 2256970375232288*x^17 + 9498116639867573*x^18 - 25485484994020128*x^19 + 37162639109810884*x^20 - 37419816866322296*x^21 + 27200926921683600*x^22 - 14055671260790656*x^23 + 4698364855901568*x^24 - 583485067952640*x^25 - 341605998065664*x^26 + 237336648708096*x^27 - 72380729917440*x^28 + 11910492979200*x^29 - 859963392000*x^30 ;
...
where the coefficient of x^(k*(k+1)) in P(k,x) equals A059332(k+1).
(End)
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PROG
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(PARI) {T(n, k) = my(A=x); for(i=1, n, A = x + subst(intformal(A +x*O(x^n)), x, y*A + (1-y)*x ) ); n!*polcoeff(polcoeff(A, n, x), k, y)}
for(n=1, 12, for(k=0, n-1, print1(T(n, k), ", ")); print(""))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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