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A277410 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. 15
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..1081, of rows n=1..46 of this triangle in flattened form.

FORMULA

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.

EXAMPLE

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.

...

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.

A210949(n) = Sum_{k=0..n-1} A277410(n,k).

A277403(n) = Sum_{k=0..n-1} A277410(n,k) * 2^(n-k-1).

A279843(n) = Sum_{k=0..n-1} A277410(n,k) * 3^(n-k-1).

A279844(n) = Sum_{k=0..n-1} A277410(n,k) * 2^k * 3^(n-k-1).

A279845(n) = Sum_{k=0..n-1} A277410(n,k) * 2^k.

A280570(n) = Sum_{k=0..n-1} A277410(n,k) * 4^(n-k-1).

A280571(n) = Sum_{k=0..n-1} A277410(n,k) * 3^k * 4^(n-k-1).

A280572(n) = Sum_{k=0..n-1} A277410(n,k) * 5^(n-k-1).

A280573(n) = Sum_{k=0..n-1} A277410(n,k) * 2^k * 5^(n-k-1).

A280574(n) = Sum_{k=0..n-1} A277410(n,k) * 3^k * 5^(n-k-1).

A280575(n) = Sum_{k=0..n-1} A277410(n,k) * 4^k * 5^(n-k-1).

...

COLUMN GENERATING FUNCTIONS.

From Paul D. Hanna, Nov 05 2016: (Start)

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)

PROG

(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(""))

CROSSREFS

Cf. A210949 (row sums), A067146, A001147 (diagonal), A277411 (column 1), A277412 (diagonal).

Cf. A279843, A279844, A279845, A280570, A280571, A280572, A280573, A280574, A280575.

Sequence in context: A245111 A135313 A322670 * A289546 A334823 A279031

Adjacent sequences:  A277407 A277408 A277409 * A277411 A277412 A277413

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Oct 13 2016

STATUS

approved

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Last modified October 1 12:45 EDT 2022. Contains 357147 sequences. (Running on oeis4.)