cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Original entry on oeis.org

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, 68098, 46558, 14373, 2881, 440, 57, 7, 1, 1122952, 744320, 221952, 42604, 6135, 728, 77, 8, 1, 21488640, 13889080, 4029915, 748548, 103326, 11565, 1120, 100
Offset: 0

Views

Author

Paul D. Hanna, Nov 18 2007

Keywords

Examples

			Triangle 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;
68098, 46558, 14373, 2881, 440, 57, 7, 1;
1122952, 744320, 221952, 42604, 6135, 728, 77, 8, 1;
21488640, 13889080, 4029915, 748548, 103326, 11565, 1120, 100, 9, 1; ...
Coefficients in iterations 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 one diagonal in the above table into another;
start with the main diagonal of A122888, A112319, which begins:
[1, 1, 2, 9, 64, 630, 7916, 121023, 2179556, 45179508, ...];
then the transform T*A112319 equals A112317, which begins:
[1, 2, 6, 30, 220, 2170, 27076, 409836, 7303164, 149837028, ...];
and the transform T*A112317 equals A112320, which begins:
[1, 3, 12, 70, 560, 5810, 74760, 1153740, 20817588, 430604724, ...].

Links

Crossrefs

Cf. columns: A135081, A135082, A135083.
Cf. related tables: A122888, A166900, A187005, A187115, A187120.
Cf. related sequences: A112319, A112317, A112320, A187009.

Programs

  • PARI
    {T(n,k)=local(F=x,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,x+x^2+x*O(x^(m+2))));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]}
  • PARI
    /* Generate by method given in A187005, A187115, A187120 (faster): */
{T(n,k)=local(Ck=x);for(m=1,n-k+1,Ck=(1/x^k)*subst(truncate(x^k*Ck),x,x+x^2 +x*O(x^m)));polcoeff(Ck,n-k+1,x)}

Formula

Columns may be generated by a method illustrated by triangles A187005, A187115, and A187120. The main diagonal of triangles A187005, A187115, and A187120, equals columns 0, 1, and 2, respectively.

Extensions

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

A135081 Column 0 of triangle A135080.

Original entry on oeis.org

1, 1, 2, 8, 50, 436, 4912, 68098, 1122952, 21488640, 468331252, 11456367820, 310888085872, 9269621420284, 301268634277760, 10601062978739338, 401550210033474420, 16291237867482727084, 704847239600911931248
Offset: 0

Views

Author

Paul D. Hanna, Nov 18 2007

Keywords

Comments

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.

Links

Crossrefs

Cf. A135080 (triangle); other columns: A135082, A135083.
Cf. A187005, A187006, A187007, A187008, A187009.

Programs

  • PARI
    /* 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]}
  • PARI
    /* 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)}

Formula

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

Views

Author

Paul D. Hanna, Nov 18 2007

Keywords

Comments

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

Crossrefs

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

Programs

  • PARI
    {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]}

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

Views

Author

Paul D. Hanna, Mar 08 2011

Keywords

Examples

			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.

Crossrefs

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

Programs

  • PARI
    {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)}
  • PARI
    {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(""))}

Formula

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.

A187124 G.f. A(x) satisfies: [x^(n+3)] 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, -3, 6, -18, 48, -195, 549, -3465, 7452, -112707, -5994, -6866904, -25659292, -700243362, -5594278734, -106900155574, -1284177510456, -22692117042216, -348993455353854, -6343625959542180, -114598750263323292, -2239367384128230334, -45442026505346961480, -967951044447512385336
Offset: 3

Views

Author

Paul D. Hanna, Mar 08 2011

Keywords

Examples

			G.f.: A(x) = x^3 - 3*x^4 + 6*x^5 - 18*x^6 + 48*x^7 - 195*x^8 +...
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, -3, 6, -18, 48, -195, 549, -3465, 7452, -112707, ...];
  [1, 0, -3, -5, -12, -72, -333, -2568, -16782, -153204, ...];
  [1, 3, 0, -19, -72, -261, -1276, -8079, -58932, -486635, ...];
  [1, 6, 15, 0, -174, -1047, -5256, -29676, -202908, -1625427, ...];
  [1, 9, 42, 112, 0, -2109, -17211, -112371, -753606, -5711283, ...];
  [1, 12, 81, 377, 1128, 0, -31633, -324600, -2614344, -20650886, ...];
  [1, 15, 132, 855, 4248, 14373, 0, -564081, -6957390, -66648777, ...];
  [1, 18, 195, 1606, 10758, 58269, 221952, 0, -11639502, -167467539,..];
  [1, 21, 270, 2690, 22416, 159633, 947117, 4029915, 0, -272551739,...];
  [1, 24, 357, 4167, 41340, 359616, 2750067, 17848872, 84135510, 0,...]; ...
in which the main diagonal equals all zeros after the initial '1';
the lower triangular portion of the above table forms triangle A187124.

Crossrefs

Cf. A187009, A187119, A187120, A135083, A187121, A187122, A187123.

Programs

  • PARI
    {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^3*Ser(A), x, ITERATE(x+x^2, i, #A)))[#A]); A[n]}

Extensions

More terms from Michel Marcus, Jan 27 2025

A187121 A diagonal of triangle A187120.

Original entry on oeis.org

1, 6, 42, 377, 4248, 58269, 947117, 17848872, 383237040, 9243654925, 247586590398, 7293962774574, 234458181733224, 8167132024738422, 306500617604837898, 12329458457556027538, 529269910501209999900
Offset: 3

Views

Author

Paul D. Hanna, Mar 08 2011

Keywords

Comments

Definition of triangle: A187120(n,k) = [y^k] R_{n-1}(y+y^2) for k=3..n, n>3, where R_n(y) is the n-th row polynomial starting with R_3(y)=y^3.

Crossrefs

Cf. A187120, A135083, A187122, A187123, A187124.

Programs

  • PARI
    {a(n)=local(Rn=y^3);for(m=3,n,Rn=subst(truncate(Rn),y,y+y^2+O(y^m)));polcoeff(Rn,n,y)}

Formula

Equals column 2 in the matrix square of triangle A135080.

A187122 A diagonal of triangle A187120.

Original entry on oeis.org

1, 9, 81, 855, 10758, 159633, 2750067, 54178485, 1204443432, 29871630837, 818490738402, 24571782872034, 802459134168208, 28332664539686670, 1075700621922471621, 43710289920461797346, 1893011243289589171122
Offset: 3

Views

Author

Paul D. Hanna, Mar 08 2011

Keywords

Comments

Definition of triangle: A187120(n,k) = [y^k] R_{n-1}(y+y^2) for k=3..n, n>3, where R_n(y) is the n-th row polynomial starting with R_3(y)=y^3.

Crossrefs

Cf. A187120, A135083, A187121, A187123, A187124.

Programs

  • PARI
    {a(n)=local(Rn=y^3);for(m=3,n+1,Rn=subst(truncate(Rn),y,y+y^2+O(y^m)));polcoeff(Rn,n,y)}

Formula

Equals column 2 in the matrix cube of triangle A135080.

A187123 Row sums of triangle A187120.

Original entry on oeis.org

1, 4, 22, 164, 1599, 19624, 292799, 5162063, 105139954, 2430528374, 62877375426, 1799698534522, 56461862815496, 1926534563625822, 71024188655886038, 2813193432059656268, 119136308527407025379, 5371507426340905440926
Offset: 3

Views

Author

Paul D. Hanna, Mar 08 2011

Keywords

Comments

Definition of triangle: A187120(n,k) = [y^k] R_{n-1}(y+y^2) for k=3..n, n>3, where R_n(y) is the n-th row polynomial starting with R_3(y)=y^3.

Crossrefs

Cf. A187120, A135083, A187121, A187122, A187124.

Programs

  • PARI
    {a(n)=local(Rn=y^3);for(m=3,n-1,Rn=subst(truncate(Rn),y,y+y^2+O(y^m)));subst(truncate(Rn),y,1)}
Showing 1-8 of 8 results.

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