A135080 -id:A135080 - cp's OEIS frontend

A135081 Column 0 of triangle A135080.

1, 1, 2, 8, 50, 436, 4912, 68098, 1122952, 21488640, 468331252, 11456367820, 310888085872, 9269621420284, 301268634277760, 10601062978739338, 401550210033474420, 16291237867482727084, 704847239600911931248

Offset: 0

Paul D. Hanna, Nov 18 2007

nonn

Triangle A135080 transforms diagonals in the table of coefficients of successive iterations of x+x^2 (cf. A122888).

Triangle A187005 is defined by: A187005(n,k) = [y^k] R_{n-1}(y+y^2) for k=1..n where R_n(y) is the n-th row polynomial for n>1 with R_1(y)=y.

Paul D. Hanna, Table of n, a(n) for n=0..50

Cf. A135080 (triangle); other columns: A135082, A135083.

Cf. A187005, A187006, A187007, A187008, A187009.

/* As column 0 of triangle A135080 (slower): */
{a(n)=local(F=x,M,N,P); M=matrix(n+2,n+2,r,c,F=x;for(i=1,r+c-2,F=subst(F,x,x+x^2+x*O(x^(n+2))));polcoeff(F,c)); N=matrix(n+1,n+1,r,c,M[r,c]);P=matrix(n+1,n+1,r,c,M[r+1,c]);(P~*N~^-1)[n+1,1]}

/* As the main diagonal of triangle A187005 (faster): */
{a(n)=local(Rn=y); for(m=1, n+1, Rn=subst(truncate(Rn), y, y+y^2+y*O(y^m))); polcoeff(Rn/y, n, y)}

Equals the main diagonal of triangle A187005.

A135082 Column 1 of triangle A135080.

Original entry on oeis.org

1, 2, 7, 40, 326, 3492, 46558, 744320, 13889080, 296459376, 7125938790, 190502850972, 5607258255032, 180198503713952, 6278311585490032, 235730921392184452, 9489040823468191328, 407662178549724426176

Offset: 0

Paul D. Hanna, Nov 18 2007

nonn

Triangle A135080 transforms diagonals in the table of coefficients of successive self-compositions of x+x^2 (cf. A122888).

Cf. A135080 (triangle); other columns: A135081, A135083.

{a(n)=local(F=x,M,N,P); M=matrix(n+3,n+3,r,c,F=x;for(i=1,r+c-2,F=subst(F,x,x+x^2+x*O(x^(n+3))));polcoeff(F,c)); N=matrix(n+2,n+2,r,c,M[r,c]);P=matrix(n+2,n+2,r,c,M[r+1,c]);(P~*N~^-1)[n+2,2]}

A135083 Column 2 of triangle A135080.

Original entry on oeis.org

1, 3, 15, 112, 1128, 14373, 221952, 4029915, 84135510, 1985740905, 52277994396, 1518768476508, 48261093246396, 1665034362336120, 61979166611850084, 2475861386988907814, 105641851808320785498, 4795101548183135826810

Offset: 0

Paul D. Hanna, Nov 18 2007

nonn

Triangle A135080 transforms diagonals in the table of coefficients of successive self-compositions of x+x^2 (cf. A122888).

Cf. A135080 (triangle); other columns: A135081, A135082.

{a(n)=local(F=x,M,N,P); M=matrix(n+4,n+4,r,c,F=x;for(i=1,r+c-2,F=subst(F,x,x+x^2+x*O(x^(n+4))));polcoeff(F,c)); N=matrix(n+3,n+3,r,c,M[r,c]);P=matrix(n+3,n+3,r,c,M[r+1,c]);(P~*N~^-1)[n+3,3]}

A112317 Coefficients of x^n in the n-th iteration of (x + x^2) for n>=1.

Original entry on oeis.org

1, 2, 6, 30, 220, 2170, 27076, 409836, 7303164, 149837028, 3479498880, 90230486346, 2584679465160, 81056989408928, 2762187020749144, 101633218030586364, 4015771398425994048, 169588657820702174728

Offset: 1

Paul D. Hanna, Sep 03 2005

nonn

Forms a diagonal of the tables A122888 and A185755.

			The initial iterations of x + x^2 begin:
F(x) = (1)*x + x^2;
F(F(x)) = x + (2)*x^2 + 2*x^3 + x^4;
F(F(F(x))) = x + 3*x^2 + (6)*x^3 + 9*x^4+ 10*x^5+ 8*x^6+ 4*x^7+ x^8;
F(F(F(F(x)))) = x + 4*x^2 + 12*x^3 + (30)*x^4 + 64*x^5 +...;
F(F(F(F(F(x))))) = x + 5*x^2 + 20*x^3 + 70*x^4 + (220)*x^5 +...;
F(F(F(F(F(F(x)))))) = x + 6*x^2 + 30*x^3 + 135*x^4 + 560*x^5 + (2170)*x^6 +...;
where the terms in parenthesis illustrate how to form this sequence.

Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 1..300 (first 100 terms from Paul D. Hanna)

Cf. A112319, A112320, A122888, A185755, A135080, A166900.

{a(n)=local(F=x+x^2, G=x+x*O(x^n));if(n<1,0, for(i=1,n,G=subst(F,x,G));return(polcoeff(G,n,x)))}
for(n=1, 30, print1(a(n), ", "))

a(n) = [x^n] F_n(x) where F_n(x) = F_{n-1}(x+x^2) with F_1(x) = x+x^2.

Added cross-references and comments; name and example changed by Paul D. Hanna, Feb 04 2011

A158835 Triangle, read by rows, that transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 27, 11, 3, 1, 254, 94, 21, 4, 1, 3062, 1072, 217, 34, 5, 1, 45052, 15212, 2904, 412, 50, 6, 1, 783151, 257777, 47337, 6325, 695, 69, 7, 1, 15712342, 5074738, 906557, 116372, 12035, 1082, 91, 8, 1, 357459042, 113775490, 19910808, 2483706

Offset: 1

Paul D. Hanna, Mar 28 2009, Mar 29 2009

Conjecture: n-th reversed row polynomial is t_n where we start with vector v of fixed length m with elements v_i = 1, then set t := v and for i=1..m-1, for j=1..i, for k=j+1..i+1 apply v_k := v_k + z*v_{k-1} and t_{i+1} := v_{i+1} (after ending each cycle for j). - Mikhail Kurkov, Sep 03 2024

			Triangle T begins:
  1;
  1,1;
  4,2,1;
  27,11,3,1;
  254,94,21,4,1;
  3062,1072,217,34,5,1;
  45052,15212,2904,412,50,6,1;
  783151,257777,47337,6325,695,69,7,1;
  15712342,5074738,906557,116372,12035,1082,91,8,1;
  357459042,113775490,19910808,2483706,246596,20859,1589,116,9,1;
  9094926988,2861365660,492818850,60168736,5801510,470928,33747,2232,144,10,1;
  ...
Array A158825 of coefficients in iterations of x*C(x) begins:
  1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,...;
  1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
  1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,...;
  1,4,20,110,640,3870,24084,153306,993978,6544242,43652340,...;
  1,5,30,195,1330,9380,67844,500619,3755156,28558484,...;
  1,6,42,315,2464,19852,163576,1372196,11682348,100707972,...;
  1,7,56,476,4200,38052,351792,3305484,31478628,303208212,...;
  1,8,72,684,6720,67620,693048,7209036,75915708,807845676,...;
  1,9,90,945,10230,113190,1273668,14528217,167607066,...;
  1,10,110,1265,14960,180510,2212188,27454218,344320262,...;
  ...
This triangle transforms diagonals of A158825 into each other:
T*A158831 = A158832; T*A158832 = A158833; T*A158833 = A158834;
where:
A158831 = [1,1,6,54,640,9380,163576,3305484,...];
A158832 = [1,2,12,110,1330,19852,351792,7209036,...];
A158833 = [1,3,20,195,2464,38052,693048,14528217,...];
A158834 = [1,4,30,315,4200,67620,1273668,27454218,...].

Paul D. Hanna, Table of n, a(n), n = 1..496 (rows 1..31).

Cf. columns: A158836, A158837, A158838, A158839, row sums: A158840.

Cf. A158825, A158831, A158832, A158833, A158834, variant: A135080.

{T(n, k)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+2))), M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

Edited by N. J. A. Sloane, Oct 04 2010, to make entries, offset, b-file and link to b-file all consistent.

A166900 Triangle, read by rows, that transforms rows into diagonals in the table of coefficients of successive iterations of x+x^2 (cf. A122888).

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 9, 21, 9, 1, 64, 156, 84, 16, 1, 630, 1540, 935, 230, 25, 1, 7916, 19160, 12480, 3564, 510, 36, 1, 121023, 288813, 196623, 61845, 10465, 987, 49, 1, 2179556, 5123608, 3591560, 1207696, 228800, 25864, 1736, 64, 1, 45179508, 104657520

Offset: 0

Paul D. Hanna, Nov 27 2009

Compare to the triangle A071207 that transforms rows into diagonals in the table of iterations of x/(1-x), where A071207(n,k) gives the number of labeled free trees with n vertices and k children of the root that have a label smaller than the label of the root. Does this triangle have a similar interpretation?

			Triangle begins:
1;
1, 1;
2, 4, 1;
9, 21, 9, 1;
64, 156, 84, 16, 1;
630, 1540, 935, 230, 25, 1;
7916, 19160, 12480, 3564, 510, 36, 1;
121023, 288813, 196623, 61845, 10465, 987, 49, 1;
2179556, 5123608, 3591560, 1207696, 228800, 25864, 1736, 64, 1;
45179508, 104657520, 74847168, 26415840, 5426949, 695079, 56511, 2844, 81, 1;
1059312264, 2420186616, 1755406674, 642448632, 140247810, 19683060, 1830080, 112520, 4410, 100, 1; ...
Coefficients in self-compositions of (x+x^2) form table A122888:
1;
1, 1;
1, 2, 2, 1;
1, 3, 6, 9, 10, 8, 4, 1;
1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...; ...
This triangle T transforms rows of A122888 into diagonals of A122888;
the initial diagonals begin:
A112319: [1, 1, 2, 9, 64, 630, 7916, 121023, 2179556, 45179508, ...];
A112317: [1, 2, 6, 30, 220, 2170, 27076, 409836, 7303164, 149837028,..];
A112320: [1, 3, 12, 70, 560, 5810, 74760, 1153740, 20817588, 430604724, ...].
For example:
T * [1, 0, 0, 0, 0, 0, 0,...]~ = A112319;
T * [1, 1, 0, 0, 0, 0, 0,...]~ = A112317;
T * [1, 2, 2, 1, 0, 0, 0,...]~ = A112320.

Paul D. Hanna, Table of n, a(n) for n=0..495 (rows 0..30)

Cf. A112319, A166901, A166902, A166903, A122888, A135080.

{T(n, k)=local(F=x, M, N, P, m=n); M=matrix(m+2, m+2, r, c, F=x;for(i=1, r+c-2, F=subst(F, x, x+x^2+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, F=x;for(i=1, r, F=subst(F, x, x+x^2+x*O(x^(m+2)))); polcoeff(F, c)); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

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.

Original entry on oeis.org

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, 1880, 4912, 1, 7, 42, 230, 1164, 5404, 22108, 68098, 1, 8, 56, 363, 2210, 12646, 67092, 315784, 1122952, 1, 9, 72, 539, 3830, 25930, 166520, 997581, 5322126, 21488640, 1

Offset: 1

Paul D. Hanna, Mar 02 2011

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

			Triangle begins:
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, 1880, 4912;
1, 7, 42, 230, 1164, 5404, 22108, 68098;
1, 8, 56, 363, 2210, 12646, 67092, 315784, 1122952;
1, 9, 72, 539, 3830, 25930, 166520, 997581, 5322126, 21488640;
1, 10, 90, 764, 6202, 48386, 362556, 2591010, 17337444, 103541022, 468331252; ...
in which rows can be generated as illustrated below.
Row polynomials R_n(y) begin:
R_1(y) = y;
R_2(y) = y + y^2;
R_3(y) = y + 2*y^2 + 2*y^3;
R_4(y) = y + 3*y^2 + 6*y^3 + 8*y^4;
R_5(y) = y + 4*y^2 + 12*y^3 + 29*y^4 + 50*y^5; ...
where row n = the coefficients of y^k in R_{n-1}(y+y^2) for k=1..n;
this method is illustrated by:
n=3: R_2(y+y^2) = (y + 2*y^2 + 2*y^3) + y^4;
n=4: R_3(y+y^2) = (y + 3*y^2 + 6*y^3 + 8*y^4) + 6*y^5 + 2*y^6;
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;
where the n-th row polynomial R_n(y) equals R_{n-1}(y+y^2) truncated to the initial n terms.
...
ALTERNATE GENERATING METHOD.
Let F^n(x) denote the n-th iteration of x+x^2 with F^0(x) = x.
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:
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 +...
and satisfies: [x^(n+1)] G(F^n(x)) = 0 for n>0.
The table of coefficients in G(F^n(x)) begins:
G(x+x^2) : [1, 0, 0, -1, 2, -14, 44, -348, 1476, -14148, ...];
G(F^2(x)): [1, 1, 0, -1, -2, -10, -24, -231, -654, -9276, ...];
G(F^3(x)): [1, 2, 2, 0, -6, -26, -108, -570, -3216, -22622, ...];
G(F^4(x)): [1, 3, 6, 8, 0, -54, -324, -1776, -10594, -71702, ...];
G(F^5(x)): [1, 4, 12, 29, 50, 0, -616, -4846, -32686, -228926, ...];
G(F^6(x)): [1, 5, 20, 69, 202, 436, 0, -8629, -84140, -680298, ...];
G(F^7(x)): [1, 6, 30, 134, 538, 1880, 4912, 0, -143442, -1672428, ..];
G(F^8(x)): [1, 7, 42, 230, 1164, 5404, 22108, 68098, 0, -2762748, ..];
G(F^9(x)): [1, 8, 56, 363, 2210, 12646, 67092, 315784, 1122952, 0, ..]; ...
of which this triangle forms the lower triangular portion.
...
TRANSFORMATIONS OF DIAGONALS BY TRIANGLE A135080.
Given main diagonal = A135081 = [1,1,2,8,50,436,4912,68098,...],
the diagonals can be generated from each other as illustrated by:
_ A135080 * A135081 = A187006 = [1,2,6,29,202,1880,22108,315784,...];
_ A135080 * A187006 = A187007 = [1,3,12,69,538,5404,67092,997581,...];
_ A135080 * A187007 = [1,4,20,134,1164,12646,166520,2591010,...].
Related triangle A135080 begins:
1;
1, 1;
2, 2, 1;
8, 7, 3, 1;
50, 40, 15, 4, 1;
436, 326, 112, 26, 5, 1;
4912, 3492, 1128, 240, 40, 6, 1; ...

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

Cf. A187009, A135080, A187115, A187120.

f[p_] := Series[p /. y -> y + y^2, {y, 0, 1 + Exponent[p, y]}] // Normal;
Flatten[ Rest[ CoefficientList[#, y]] & /@ NestList[f, y, 10]][[1 ;; 56]] (* Jean-François Alcover, Jun 09 2011 *)

{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)}
for(n=1,10,for(k=1,n,print1(T(n,k),", "));print(""))

{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))))}
for(n=1,10,for(k=1,n,print1(T(n,k),", "));print(""))

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.
Main diagonal equals column 0 of triangle A135080, which transforms diagonals in the table of coefficients of the iterations of x+x^2.
Triangle A135080 also transforms diagonals in this triangle into each other.
Diagonal m of this triangle equals column 0 of the m-th power of triangle A135080, with diagonal m=1 being the main diagonal.

A187009 G.f. A(x) satisfies: [x^(n+1)] A(F^n(x)) = 0 for n>0 where F^n(x) denotes the n-th iteration of F(x) = x+x^2 with F^0(x)=x.

Original entry on oeis.org

1, -1, 2, -6, 20, -80, 348, -1778, 9892, -64392, 449596, -3609782, 30152616, -284037468, 2694480888, -28592860322, 295151311376, -3440953545088, 37165311149276, -471576198145144, 5062381083026352, -71104461751595892

Offset: 1

Paul D. Hanna, Mar 02 2011

sign

			G.f.: A(x) = x - x^2 + 2*x^3 - 6*x^4 + 20*x^5 - 80*x^6 + 348*x^7 +...
Let F^n(x) denote the n-th iteration of F(x) = x+x^2 with F^0(x)=x,
then the table of coefficients in A(F^n(x)), n>=0, begins:
[1, -1, 2, -6, 20, -80, 348, -1778, 9892, -64392, ...];
[1, 0, 0, -1, 2, -14, 44, -348, 1476, -14148, 73920, ...];
[1, 1, 0, -1, -2, -10, -24, -231, -654, -9276, -32456, ...];
[1, 2, 2, 0, -6, -26, -108, -570, -3216, -22622, -162596, ...];
[1, 3, 6, 8, 0, -54, -324, -1776, -10594, -71702, -540448, ...];
[1, 4, 12, 29, 50, 0, -616, -4846, -32686, -228926, -1749972, ...];
[1, 5, 20, 69, 202, 436, 0, -8629, -84140, -680298, -5508864, ...];
[1, 6, 30, 134, 538, 1880, 4912, 0, -143442, -1672428, -15821492, ...];
[1, 7, 42, 230, 1164, 5404, 22108, 68098, 0, -2762748, -37526484, ...];
[1, 8, 56, 363, 2210, 12646, 67092, 315784, 1122952, 0, -60534272, ..];
[1, 9, 72, 539, 3830, 25930, 166520, 997581, 5322126, 21488640, 0, ..]; ...
in which the main diagonal equals all zeros after the initial '1';
the lower triangular portion of the above table forms triangle A187005.

Cf. A187005, A135080, A122888, A135081, A187006, A187007, A187008.

{ITERATE(F,n,p)=local(G=x);for(i=1,n,G=subst(F,x,G+x*O(x^p)));G}
{a(n)=local(A=[1]);for(i=1,n,A=concat(A,0);A[#A]=-Vec(subst(x*Ser(A),x,ITERATE(x+x^2,i,#A)))[#A]);A[n]}

A187115 Triangle, read by rows, where row n equals the coefficients of y^k in R_{n-1}(y+y^2) for k=2..n, where R_n(y) is the n-th row polynomial in y for n>=2 with R_2(y)=y^2.

Original entry on oeis.org

1, 1, 2, 1, 4, 7, 1, 6, 20, 40, 1, 8, 39, 138, 326, 1, 10, 64, 318, 1258, 3492, 1, 12, 95, 604, 3242, 14476, 46558, 1, 14, 132, 1020, 6844, 40348, 202655, 744320, 1, 16, 175, 1590, 12750, 92140, 598083, 3354848, 13889080, 1, 18, 224, 2338, 21766, 185240, 1450388, 10337402, 64246776, 296459376

Offset: 2

Paul D. Hanna, Mar 07 2011

			Triangle begins:
1;
1, 2;
1, 4, 7;
1, 6, 20, 40;
1, 8, 39, 138, 326;
1, 10, 64, 318, 1258, 3492;
1, 12, 95, 604, 3242, 14476, 46558;
1, 14, 132, 1020, 6844, 40348, 202655, 744320;
1, 16, 175, 1590, 12750, 92140, 598083, 3354848, 13889080;
1, 18, 224, 2338, 21766, 185240, 1450388, 10337402, 64246776, 296459376;
1, 20, 279, 3288, 34818, 340112, 3097162, 26277556, 204706486, 1398909332, 7125938790; ...
in which rows can be generated as illustrated below.
Row polynomials R_n(y), n>=2, begin:
R_2(y) = y^2;
R_3(y) = y^2 + 2*y^3;
R_4(y) = y^2 + 4*y^3 + 7*y^4;
R_5(y) = y^2 + 6*y^3 + 20*y^4 + 40*y^5;
R_6(y) = y^2 + 8*y^3 + 39*y^4 + 138*y^5 + 326*y^6; ...
where row n = coefficients of y^k in R_{n-1}(y+y^2) for k=2..n;
this method is illustrated by:
n=3: R_2(y+y^2) = (y^2 + 2*y^3) + y^4;
n=4: R_3(y+y^2) = (y^2 + 4*y^3 + 7*y^4) + 6*y^5 + 2*y^6;
n=5: R_4(y+y^2) = (y^2 + 6*y^3 + 20*y^4 + 40*y^5) + 46*y^6 + 28*y^7 + 7*y^8;
n=6: R_5(y+y^2) = (y^2 + 8*y^3 + 39*y^4 + 138*y^5 + 326*y^6) + 480*y^7 + 420*y^8 + 200*y^9 + 40*y^10;
where the n-th row polynomial R_n(y) equals R_{n-1}(y+y^2) truncated to the initial n-1 nonzero terms.
...
ALTERNATE GENERATING METHOD.
Let F^n(x) denote the n-th iteration of x+x^2 with F^0(x) = x.
Then row n of this triangle may be generated by the coefficients of x^k in G(F^n(x)), k=2..n, n>=2, where G(x) is the g.f. of A187119:
G(x) = x^2 - 2*x^3 + 4*x^4 - 12*x^5 + 36*x^6 - 140*x^7 + 519*x^8 - 2632*x^9 + 11776*x^10 - 82020*x^11 + 426990*x^12 +...
and satisfies: [x^(n+2)] G(F^n(x)) = 0 for n>0.
The table of coefficients in G(F^n(x)) begins:
G(x+x^2) : [1, 0, -1, -2, -2, -28, -37, -760, -1752,...];
G(F^2(x)): [1, 2, 0, -6, -18, -64, -284, -1694, -10640, ...];
G(F^3(x)): [1, 4, 7, 0, -46, -232, -1062, -5700, -36354, ...];
G(F^4(x)): [1, 6, 20, 40, 0, -480, -3369, -19988, -126200, ...];
G(F^5(x)): [1, 8, 39, 138, 326, 0, -6309, -56820, -417184, ...];
G(F^6(x)): [1, 10, 64, 318, 1258, 3492, 0, -100082, -1100188, ...];
G(F^7(x)): [1, 12, 95, 604, 3242, 14476, 46558, 0, -1859518, ...];
G(F^8(x)): [1, 14, 132, 1020, 6844, 40348, 202655, 744320, 0, ...];
of which this triangle forms the lower triangular portion.
...
TRANSFORMATIONS OF SHIFTED DIAGONALS BY TRIANGLE A135080.
Given main diagonal = A135082 = [0,1,2,7,40,326,3492,46558,...],
the diagonals can be generated from each other as illustrated by:
_ A135080 * A135082 = A187116 = [0,1,4,20,138,1258,14476,202655,...];
_ A135080 * A187116 = A187117 = [0,1,6,39,318,3242,40348,598083,...];
_ A135080 * A187117 = [0,1,8,64,604,6844,92140,1450388,...],
where a leading zero is included in forming the vectors.
Related triangle A135080 begins:
1;
1, 1;
2, 2, 1;
8, 7, 3, 1;
50, 40, 15, 4, 1;
436, 326, 112, 26, 5, 1;
4912, 3492, 1128, 240, 40, 6, 1; ...
where column 1 of A135080 is the main diagonal in this triangle.

Cf. diagonals: A135082, A187116, A187117; row sums: A187118.
Cf. related triangles: A135080, A187005, A187120.
Cf. A187119.

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

{T(n,k)=if(k>n||k<2,0,if(n==2,1,sum(j=k\2,k,binomial(j,k-j)*T(n-1,j))))}
/* Print the triangle: */
{for(n=2,12,for(k=2,n,print1(T(n,k),","));print(""))}

T(n,k) = Sum_{j=[k/2],k} C(j,k-j)*T(n-1,j) for n>=2, k=2..n, with T(n,2)=1 and T(n,k)=0 when k>n or k<2.
Main diagonal equals column 1 of triangle A135080, which transforms diagonals in the table of coefficients of the iterations of x+x^2.
Triangle A135080 also transforms diagonals in this triangle into each other.
Diagonal m of this triangle equals column 1 of the m-th power of triangle A135080, with diagonal m=1 being the main diagonal.

A187120 Triangle, read by rows, where row n equals the coefficients of y^k in R_{n-1}(y+y^2) for k=3..n, where R_n(y) is the n-th row polynomial in y for n>=3 with R_3(y)=y^3.

Original entry on oeis.org

1, 1, 3, 1, 6, 15, 1, 9, 42, 112, 1, 12, 81, 377, 1128, 1, 15, 132, 855, 4248, 14373, 1, 18, 195, 1606, 10758, 58269, 221952, 1, 21, 270, 2690, 22416, 159633, 947117, 4029915, 1, 24, 357, 4167, 41340, 359616, 2750067, 17848872, 84135510, 1, 27, 456, 6097

Offset: 3

Paul D. Hanna, Mar 08 2011

			Triangle begins:
1;
1, 3;
1, 6, 15;
1, 9, 42, 112;
1, 12, 81, 377, 1128;
1, 15, 132, 855, 4248, 14373;
1, 18, 195, 1606, 10758, 58269, 221952;
1, 21, 270, 2690, 22416, 159633, 947117, 4029915;
1, 24, 357, 4167, 41340, 359616, 2750067, 17848872, 84135510;
1, 27, 456, 6097, 70008, 715095, 6580260, 54178485, 383237040, 1985740905;
1, 30, 567, 8540, 111258, 1301193, 13895408, 135965676, 1204443432, 9243654925, 52277994396; ...
in which rows can be generated as illustrated below.
Row polynomials R_n(y), n>=3, begin:
R_3(y) = y^3;
R_4(y) = y^3 + 3*y^4;
R_5(y) = y^3 + 6*y^4 + 15*y^5;
R_6(y) = y^3 + 9*y^4 + 42*y^5 + 112*y^6;
R_7(y) = y^3 + 12*y^4 + 81*y^5 + 377*y^6 + 1128*y^7; ...
where row n = coefficients of y^k in R_{n-1}(y+y^2) for k=3..n;
this method is illustrated by:
n=4: R_3(y+y^2) = (y^3 + 3*y^4) + 3*y^5 + y^6;
n=5: R_4(y+y^2) = (y^3 + 6*y^4 + 15*y^5) + 19*y^6 + 12*y^7 + 3*y^8;
n=6: R_5(y+y^2) = (y^3 + 9*y^4 + 42*y^5 + 112*y^6) + 174*y^7 + 156*y^8 + 75*y^9 + 15*y^10; ...
where the n-th row polynomial R_n(y) equals R_{n-1}(y+y^2) truncated to the initial n-2 nonzero terms.
...
ALTERNATE GENERATING METHOD.
Let F^n(x) denote the n-th iteration of x+x^2 with F^0(x) = x.
Then row n of this triangle may be generated by the coefficients of x^k in G(F^[n-2](x)), k=3..n, n>=3, where G(x) is the g.f. of A187124:
G(x) = x^3 - 3*x^4 + 6*x^5 - 18*x^6 + 48*x^7 - 195*x^8 + 549*x^9 - 3465*x^10 + 7452*x^11 - 112707*x^12 - 5994*x^13 - 6866904*x^14 +...
and satisfies: [x^(n+2)] G(F^n(x)) = 0 for n>0.
The table of coefficients in G(F^n(x)) begins:
G(x+x^2) : [1, 0, -3, -5, -12, -72, -333, -2568, -16782, ...];
G(F^2(x)): [1, 3, 0, -19, -72, -261, -1276, -8079, -58932, ...];
G(F^3(x)): [1, 6, 15, 0, -174, -1047, -5256, -29676, -202908, ...];
G(F^4(x)): [1, 9, 42, 112, 0, -2109, -17211, -112371, -753606, ...];
G(F^5(x)): [1, 12, 81, 377, 1128, 0, -31633, -324600, -2614344, ...];
G(F^6(x)): [1, 15, 132, 855, 4248, 14373, 0, -564081, -6957390, ...];
G(F^7(x)): [1, 18, 195, 1606, 10758, 58269, 221952, 0, -11639502,..];
G(F^8(x)): [1, 21, 270, 2690, 22416, 159633, 947117, 4029915, 0,...]; ...
of which this triangle forms the lower triangular portion.
...
TRANSFORMATIONS OF SHIFTED DIAGONALS BY TRIANGLE A135080.
Given main diagonal = A135083 = [0,0,1,3,15,112,1128,14373,...],
the diagonals can be generated from each other as illustrated by:
_ A135080 * A135083 = A187121 = [0,0,1,6,42,377,4248,58269,...];
_ A135080 * A187121 = A187122 = [0,0,1,9,81,855,10758,159633,...];
_ A135080 * A187122 = [0,0,1,12,132,1606,22416,359616,...],
where two leading zeros are included in forming the vectors.
Related triangle A135080 begins:
1;
1, 1;
2, 2, 1;
8, 7, 3, 1;
50, 40, 15, 4, 1;
436, 326, 112, 26, 5, 1;
4912, 3492, 1128, 240, 40, 6, 1; ...
where column 2 of A135080 is the main diagonal in this triangle.

Cf. diagonals: A135083, A187121, A187122; row sums: A187123.
Cf. related triangles: A135080, A187005, A187115.
Cf. A187124.

{T(n,k)=local(Rn=y^3);for(m=3,n-1,Rn=subst(truncate(Rn),y,y+y^2+O(y^m)));polcoeff(Rn,k,y)}

{T(n,k)=if(k>n||k<3,0,if(n==3,1,sum(j=k\2,k,binomial(j,k-j)*T(n-1,j))))}
/* Print the triangle: */
{for(n=3,13,for(k=3,n,print1(T(n,k),","));print(""))}

T(n,k) = Sum_{j=[k/2],k} C(j,k-j)*T(n-1,j) for n>=3, k=3..n, with T(n,3)=1 and T(n,k)=0 when k>n or k<3.
Main diagonal equals column 2 of triangle A135080, which transforms diagonals in the table of coefficients of the iterations of x+x^2.
Triangle A135080 also transforms diagonals in this triangle into each other.
Diagonal m of this triangle equals column 2 of the m-th power of triangle A135080, with diagonal m=1 being the main diagonal.

cp's OEIS Frontend

A135081 Column 0 of triangle A135080.

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A135082 Column 1 of triangle A135080.

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A135083 Column 2 of triangle A135080.

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A112317 Coefficients of x^n in the n-th iteration of (x + x^2) for n>=1.

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A158835 Triangle, read by rows, that transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

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A166900 Triangle, read by rows, that transforms rows into diagonals in the table of coefficients of successive iterations of x+x^2 (cf. A122888).

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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.

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A187009 G.f. A(x) satisfies: [x^(n+1)] A(F^n(x)) = 0 for n>0 where F^n(x) denotes the n-th iteration of F(x) = x+x^2 with F^0(x)=x.

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