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.

A187008 Row sums of triangle A187005.

Original entry on oeis.org

1, 2, 5, 18, 96, 733, 7501, 97054, 1521112, 28005248, 592218737, 14141289480, 376251253450, 11036022346816, 353747961265089, 12301349824260074, 461216257715290976, 18545829116907146812, 796122011317944176206
Offset: 1

Views

Author

Paul D. Hanna, Mar 02 2011

Keywords

Comments

Definition of triangle: 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.

Crossrefs

Cf. A187005, A135081, A187006, A187007, A187009.

A187006 A diagonal of triangle A187005.

Original entry on oeis.org

1, 2, 6, 29, 202, 1880, 22108, 315784, 5322126, 103541022, 2285965792, 56501970700, 1546339364952, 46433615292128, 1518172222889000, 53695946069029290, 2042960241832903824, 83207255745283689726, 3612360848988984098484, 166537270023045091100852, 8125771150231992039148508
Offset: 1

Views

Author

Paul D. Hanna, Mar 02 2011

Keywords

Comments

Definition of triangle: 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.

Crossrefs

Cf. A187005, A135081, A187007, A187008, A187009.

Programs

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

Extensions

More terms from Michel Marcus, Feb 01 2025

A187007 A diagonal of triangle A187005.

Original entry on oeis.org

1, 3, 12, 69, 538, 5404, 67092, 997581, 17337444, 345547750, 7778860028, 195365725310, 5418540898800, 164556820433116, 5432639292242576, 193765826638479112, 7426524928340527902, 304439910637309354106, 13293321078232321952036, 616016966059346344408690, 30196821383142448481822800
Offset: 1

Views

Author

Paul D. Hanna, Mar 02 2011

Keywords

Comments

Definition of triangle: 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.

Crossrefs

Cf. A187005, A135081, A187006, A187008, A187009.

Programs

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

Extensions

More terms from Michel Marcus, Feb 01 2025

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.

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

Views

Author

Paul D. Hanna, Mar 02 2011

Keywords

Examples

			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.

Crossrefs

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

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*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

Views

Author

Paul D. Hanna, Mar 07 2011

Keywords

Examples

			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.

Crossrefs

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

Programs

  • PARI
    {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)}
  • PARI
    {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(""))}

Formula

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

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.
Showing 1-8 of 8 results.

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