login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A187005 Triangle, read by rows, where row n equals the coefficients of y^k in R_{n-1}(y+y^2) for k=1..n where R_n(y) is the n-th row polynomial in y for n>1 with R_1(y)=y. 8

%I

%S 1,1,1,1,2,2,1,3,6,8,1,4,12,29,50,1,5,20,69,202,436,1,6,30,134,538,

%T 1880,4912,1,7,42,230,1164,5404,22108,68098,1,8,56,363,2210,12646,

%U 67092,315784,1122952,1,9,72,539,3830,25930,166520,997581,5322126,21488640,1

%N Triangle, read by rows, where row n equals the coefficients of y^k in R_{n-1}(y+y^2) for k=1..n where R_n(y) is the n-th row polynomial in y for n>1 with R_1(y)=y.

%C Triangles A187115 and A187120 are generated by a similar method, and have main diagonals that are also found in triangle A135080.

%F T(n,k) = Sum_{j=[k/2],k} C(j,k-j)*T(n-1,j) for n>=k>1 with T(n,1)=1 and T(n,k)=0 when k>n or k<1.

%F Main diagonal equals column 0 of triangle A135080, which transforms diagonals in the table of coefficients of the iterations of x+x^2.

%F Triangle A135080 also transforms diagonals in this triangle into each other.

%F Diagonal m of this triangle equals column 0 of the m-th power of triangle A135080, with diagonal m=1 being the main diagonal.

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 2, 2;

%e 1, 3, 6, 8;

%e 1, 4, 12, 29, 50;

%e 1, 5, 20, 69, 202, 436;

%e 1, 6, 30, 134, 538, 1880, 4912;

%e 1, 7, 42, 230, 1164, 5404, 22108, 68098;

%e 1, 8, 56, 363, 2210, 12646, 67092, 315784, 1122952;

%e 1, 9, 72, 539, 3830, 25930, 166520, 997581, 5322126, 21488640;

%e 1, 10, 90, 764, 6202, 48386, 362556, 2591010, 17337444, 103541022, 468331252; ...

%e in which rows can be generated as illustrated below.

%e Row polynomials R_n(y) begin:

%e R_1(y) = y;

%e R_2(y) = y + y^2;

%e R_3(y) = y + 2*y^2 + 2*y^3;

%e R_4(y) = y + 3*y^2 + 6*y^3 + 8*y^4;

%e R_5(y) = y + 4*y^2 + 12*y^3 + 29*y^4 + 50*y^5; ...

%e where row n = the coefficients of y^k in R_{n-1}(y+y^2) for k=1..n;

%e this method is illustrated by:

%e n=3: R_2(y+y^2) = (y + 2*y^2 + 2*y^3) + y^4;

%e n=4: R_3(y+y^2) = (y + 3*y^2 + 6*y^3 + 8*y^4) + 6*y^5 + 2*y^6;

%e n=5: R_4(y+y^2) = (y + 4*y^2 + 12*y^3 + 29*y^4 + 50*y^5) + 54*y^6 + 32*y^7 + 8*y^8;

%e where the n-th row polynomial R_n(y) equals R_{n-1}(y+y^2) truncated to the initial n terms.

%e ...

%e ALTERNATE GENERATING METHOD.

%e Let F^n(x) denote the n-th iteration of x+x^2 with F^0(x) = x.

%e Then row n of this triangle may be generated by the coefficients of x^k in G(F^n(x)), k=1..n, where G(x) is the g.f. of A187009:

%e G(x) = x - x^2 + 2*x^3 - 6*x^4 + 20*x^5 - 80*x^6 + 348*x^7 - 1778*x^8 + 9892*x^9 - 64392*x^10 + 449596*x^11 + 15449192*x^12 +...

%e and satisfies: [x^(n+1)] G(F^n(x)) = 0 for n>0.

%e The table of coefficients in G(F^n(x)) begins:

%e G(x+x^2) : [1, 0, 0, -1, 2, -14, 44, -348, 1476, -14148, ...];

%e G(F^2(x)): [1, 1, 0, -1, -2, -10, -24, -231, -654, -9276, ...];

%e G(F^3(x)): [1, 2, 2, 0, -6, -26, -108, -570, -3216, -22622, ...];

%e G(F^4(x)): [1, 3, 6, 8, 0, -54, -324, -1776, -10594, -71702, ...];

%e G(F^5(x)): [1, 4, 12, 29, 50, 0, -616, -4846, -32686, -228926, ...];

%e G(F^6(x)): [1, 5, 20, 69, 202, 436, 0, -8629, -84140, -680298, ...];

%e G(F^7(x)): [1, 6, 30, 134, 538, 1880, 4912, 0, -143442, -1672428, ..];

%e G(F^8(x)): [1, 7, 42, 230, 1164, 5404, 22108, 68098, 0, -2762748, ..];

%e G(F^9(x)): [1, 8, 56, 363, 2210, 12646, 67092, 315784, 1122952, 0, ..]; ...

%e of which this triangle forms the lower triangular portion.

%e ...

%e TRANSFORMATIONS OF DIAGONALS BY TRIANGLE A135080.

%e Given main diagonal = A135081 = [1,1,2,8,50,436,4912,68098,...],

%e the diagonals can be generated from each other as illustrated by:

%e _ A135080 * A135081 = A187006 = [1,2,6,29,202,1880,22108,315784,...];

%e _ A135080 * A187006 = A187007 = [1,3,12,69,538,5404,67092,997581,...];

%e _ A135080 * A187007 = [1,4,20,134,1164,12646,166520,2591010,...].

%e Related triangle A135080 begins:

%e 1;

%e 1, 1;

%e 2, 2, 1;

%e 8, 7, 3, 1;

%e 50, 40, 15, 4, 1;

%e 436, 326, 112, 26, 5, 1;

%e 4912, 3492, 1128, 240, 40, 6, 1; ...

%t f[p_] := Series[p /. y -> y + y^2, {y, 0, 1 + Exponent[p, y]}] // Normal;

%t Flatten[ Rest[ CoefficientList[#, y]] & /@ NestList[f, y, 10]][[1 ;; 56]] (* _Jean-Fran├žois Alcover_, Jun 09 2011 *)

%o (PARI) {T(n,k)=local(Rn=y);for(m=1,n,Rn=subst(truncate(Rn),y,y+y^2+y*O(y^m)));polcoeff(Rn,k,y)}

%o for(n=1,10,for(k=1,n,print1(T(n,k),", "));print(""))

%o (PARI) {T(n,k)=if(k>n|k<1,0,if(n==1,1,sum(j=k\2,k,binomial(j,k-j)*T(n-1,j))))}

%o for(n=1,10,for(k=1,n,print1(T(n,k),", "));print(""))

%Y Cf. diagonals: A135081, A187006, A187007; row sums: A187008.

%Y Cf. A187009, A135080, A187115, A187120.

%K nonn,tabl

%O 1,5

%A _Paul D. Hanna_, Mar 02 2011

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 11 20:03 EDT 2020. Contains 335652 sequences. (Running on oeis4.)