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.

Previous Showing 11-17 of 17 results.

A124546 Row 6 of rectangular table A124540; equals the self-convolution 6th power of A124536 (row 6 of table A124530).

Original entry on oeis.org

1, 6, 57, 470, 3756, 29658, 233241, 1836912, 14543877, 116087596, 936035298, 7636193394, 63106764294, 528842660346, 4497737044197, 38849799300246, 341016182672523, 3043519729680600, 27629723055323671, 255224042883932790
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2006

Keywords

Comments

In table A124540, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = [ Sum_{k>=0} y^k*R_k(y)^n ]^n for n>=0.

Crossrefs

Cf. A124535; A124540 (table); other rows: A124531, A124542, A124543, A124544, A124545.

Programs

  • PARI
    {a(n)=local(R);R=vector(n+7,r,vector(n+7,c,1)); for(i=0,n+6,for(r=0,n+6,R[r+1]=Vec(sum(c=0,n,x^c*Ser(R[c+1])^(r*c)+O(x^(n+1)))))); Vec(Ser(R[7])^6+O(x^(n+1)))[n+1]}

Formula

G.f.: A(x) = [ Sum_{n>=0} x^n*R_n(x)^6 ]^6, where R_n(x) is the g.f. of row n in table A124540.

A124537 Main diagonal of rectangular table A124530.

Original entry on oeis.org

1, 1, 3, 16, 113, 961, 9430, 104028, 1267833, 16866694, 242836861, 3758663745, 62200007243, 1095222618881, 20433071584276, 402407118276836, 8338150233939377, 181250019817152061, 4122472220802095509
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2006

Keywords

Comments

In table A124530, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k*R_k(y)^(n*k) for n>=0.

Crossrefs

Cf. A124530 (table).

Programs

  • PARI
    a(n)=local(R);R=vector(n+1,r,vector(n+1,c,if(r==1 || c<=2,1,r^(c-2)))); for(i=0,n,for(r=0,n,R[r+1]=Vec(sum(c=0,n,x^c*Ser(R[c+1])^(r*c)+O(x^(n+1)))))); R[n+1][n+1]

A124538 Antidiagonal sums of rectangular table A124530.

Original entry on oeis.org

1, 2, 3, 5, 11, 33, 125, 551, 2695, 14258, 80343, 477161, 2964299, 19155014, 128203704, 885830074, 6302476880, 46077527307, 345590801495, 2655466079629, 20880388012659, 167861200630015, 1378589193074514, 11558511053625488
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2006

Keywords

Comments

In table A124530, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k*R_k(y)^(n*k) for n>=0.

Crossrefs

Cf. A124530 (table).

Programs

  • PARI
    a(n)=local(R);R=vector(n+1,r,vector(n+1,c,if(r==1 || c<=2,1,r^(c-2)))); for(i=0,n,for(r=0,n,R[r+1]=Vec(sum(c=0,n,x^c*Ser(R[c+1])^(r*c)+O(x^(n+1)))))); sum(k=0,n,R[n-k+1][k+1])

A124539 Triangle, read by rows, where row n equals the inverse binomial transform of column n in the rectangular table A124530.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 15, 8, 1, 0, 1, 61, 51, 14, 1, 0, 1, 273, 311, 138, 24, 1, 0, 1, 1331, 1901, 1191, 349, 42, 1, 0, 1, 6977, 11838, 9693, 4100, 868, 76, 1, 0, 1, 38872, 75556, 76950, 43257, 13459, 2163, 142, 1, 0, 1, 228089, 495146, 606275, 430517
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2006

Keywords

Comments

In table A124530, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k*R_k(y)^(n*k) for n>=0.

Examples

			Triangle begins:
1;
1, 0;
1, 1, 0;
1, 4, 1, 0;
1, 15, 8, 1, 0;
1, 61, 51, 14, 1, 0;
1, 273, 311, 138, 24, 1, 0;
1, 1331, 1901, 1191, 349, 42, 1, 0;
1, 6977, 11838, 9693, 4100, 868, 76, 1, 0;
1, 38872, 75556, 76950, 43257, 13459, 2163, 142, 1, 0;
1, 228089, 495146, 606275, 430517, 180000, 43274, 5442, 272, 1, 0; ...

Crossrefs

Cf. A124530 (table).

Programs

  • PARI
    T(n,k)=local(m=max(n,k),R);R=vector(m+1,r,vector(m+1,c,if(r==1 || c<=2,1,r^(c-2)))); for(i=0,m, for(r=0,m, R[r+1]=Vec(sum(c=0,m, x^c*Ser(R[c+1])^(r*c)+O(x^(m+1)))))); Vec(subst(Ser(vector(n+1, j, R[j][n+1])), x, x/(1+x))/(1+x))[k+1]

Formula

Secondary diagonal T(n+1,n) = 2^n + 2n.

A124550 Rectangular table, read by antidiagonals, such that the g.f. of row n, R_n(y), satisfies: R_n(y) = [ Sum_{k>=0} y^k * R_{n*k}(y) ]^n for n>=0, with R_0(y)=1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 5, 0, 1, 4, 15, 30, 16, 0, 1, 5, 26, 91, 159, 66, 0, 1, 6, 40, 204, 666, 1056, 348, 0, 1, 7, 57, 385, 1899, 5955, 8812, 2321, 0, 1, 8, 77, 650, 4345, 21180, 65794, 92062, 19437, 0, 1, 9, 100, 1015, 8616, 57876, 287568, 901881
Offset: 0

Views

Author

Paul D. Hanna, Nov 07 2006

Keywords

Comments

Antidiagonal sums equal row 1 (A124551).

Examples

			The g.f. of row n, R_n(y), simultaneously satisfies:
R_n(y) = [1 + y*R_{n}(y) + y^2*R_{2n}(y) + y^3*R_{3n}(y) +...]^n
more explicitly,
R_0 = [1 + y + y^2 + y^3 +... ]^0 = 1,
R_1 = [1 + y*R_1 + y^2*R_2 + y^3*R_3 + y^4*R_4 +...]^1,
R_2 = [1 + y*R_2 + y^2*R_4 + y^3*R_6 + y^4*R_8 +...]^2,
R_3 = [1 + y*R_3 + y^2*R_6 + y^3*R_9 + y^4*R_12 +...]^3,
R_4 = [1 + y*R_4 + y^2*R_8 + y^3*R_12 + y^4*R_16 +...]^4,
etc., for all rows.
Table begins:
1,0,0,0,0,0,0,0,0,0,...
1,1,2,5,16,66,348,2321,19437,203554,2661035,43399794,883165898,...
1,2,7,30,159,1056,8812,92062,1200415,19512990,395379699,9991017068,...
1,3,15,91,666,5955,65794,901881,15346419,324465907,8535776700,...
1,4,26,204,1899,21180,287568,4802716,99084889,2531896840,...
1,5,40,385,4345,57876,926340,18088835,434349525,12879458545,...
1,6,57,650,8616,133212,2447115,54419202,1481595429,49675372516,...
1,7,77,1015,15449,271677,5621371,139777303,4236941723,157754261392,...
1,8,100,1496,25706,506376,11637540,319211576,10629219251,...
1,9,126,2109,40374,880326,22228296,665618589,24097683942,...
1,10,155,2870,60565,1447752,39814650,1290831110,50395939380,...
1,11,187,3795,87516,2275383,67666852,2359273213,98672395096,...
1,12,222,4900,122589,3443748,110082100,4104444564,182882370066,...
1,13,260,6201,167271,5048472,172579056,6848496031,323591733868,...
1,14,301,7714,223174,7201572,262109169,11025158762,550236760920,...
1,15,345,9455,292035,10032753,387284805,17206288875,903909656190,...
1,16,392,11440,375716,13690704,558624184,26132289904,1440743993738,...
1,17,442,13685,476204,18344394,788813124,38746675145,2235979092419,...
1,18,495,16206,595611,24184368,1092983592,56235032046,3388787136045,...
1,19,551,19019,736174,31424043,1489009062,80068650785,5027951628273,...
1,20,610,22140,900255,40301004,1997816680,112053079180,7318490555455,...
1,21,672,25585,1090341,51078300,2643716236,154381866075,10469322413655,..
1,22,737,29370,1309044,64045740,3454745943,209695755346,14742078039007,..
1,23,805,33511,1559101,79521189,4463035023,281147592671,20461165963557,..
1,24,876,38024,1843374,97851864,5705183100,372473207208,28025203801701,..

Crossrefs

Rows: A124551, A124552, A124553, A124554, A124555, A124556; diagonals: A124557, A124558, A124559; variants: A124560, A124460, A124530, A124540.

Programs

  • PARI
    {T(n,k)=if(k==0,1,if(n==0,0,if(k==1,n,if(n<=k, Vec(( 1+x*Ser( vector(k,j,sum(i=0,j-1,T(n+i*n,j-1-i)) ) ))^n)[k+1], Vec(subst(Ser(concat(concat(0, Vec(subst(Ser(vector(k+1,j,T(j-1,k))),x,x/(1+x))/(1+x))),vector(n-k+1)) ),x,x/(1-x))/(1-x +x*O(x^(n))))[n]))))}

Formula

Let G_n(y) be the g.f. of row n in table A124560, then R_n(y) = G_n(y)^n and thus G_n(y) = Sum_{k>=0} y^k * R_{n*k}(y) for n>=0, where R_n(y) is the g.f. of row n in this table.

A124560 Rectangular table, read by antidiagonals, such that the g.f. of row n, R_n(y), satisfies: R_n(y) = Sum_{k>=0} y^k * [R_{n*k}(y)]^(n*k) for n>=0, with R_0(y)=1/(1-y).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 12, 16, 1, 1, 1, 5, 22, 63, 66, 1, 1, 1, 6, 35, 158, 429, 348, 1, 1, 1, 7, 51, 317, 1455, 3716, 2321, 1, 1, 1, 8, 70, 556, 3634, 16918, 40272, 19437, 1, 1, 1, 9, 92, 891, 7581, 52199, 244644, 541655, 203554, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 07 2006

Keywords

Examples

			The g.f. of row n, R_n(y), simultaneously satisfies:
R_n(y) = 1 + y*R_{n}(y)^n + y^2*R_{2n}(y)^(2n) + y^3*R_{3n}(y)^(3n) +...
more explicitly,
R_0 = 1 + y + y^2 + y^3 +... = 1/(1-y),
R_1 = 1 + y*(R_1)^1 + y^2*(R_2)^2 + y^3*(R_3)^3 + y^4*(R_4)^4 +...,
R_2 = 1 + y*(R_2)^2 + y^2*(R_4)^4 + y^3*(R_6)^6 + y^4*(R_8)^8 +...,
R_3 = 1 + y*(R_3)^3 + y^2*(R_6)^6 + y^3*(R_9)^9 + y^4*(R_12)^12 +...,
R_4 = 1 + y*(R_4)^4 + y^2*(R_8)^8 + y^3*(R_12)^12 + y^4*(R_16)^16 +...,
etc., for all rows.
Table begins:
1,1,1,1,1,1,1,1,1,1,...
1,1,2,5,16,66,348,2321,19437,203554,2661035,43399794,883165898,...
1,1,3,12,63,429,3716,40272,541655,9022405,186233087,4771577072,...
1,1,4,22,158,1455,16918,244644,4361883,95746603,2592416878,...
1,1,5,35,317,3634,52199,928608,20282765,543008771,17866390922,...
1,1,6,51,556,7581,128532,2689248,68880819,2155007000,82603481941,...
1,1,7,70,891,14036,272914,6525900,190604859,6781448755,...
1,1,8,92,1338,23864,521662,13975298,456468525,18121964864,...
1,1,9,117,1913,38055,921709,27263527,981599065,42880525630,...
1,1,10,145,2632,57724,1531900,49474783,1941904513,92344174075,...
1,1,11,176,3511,84111,2424288,84736940,3594121407,184465174294,...
1,1,12,210,4566,118581,3685430,138423924,6299505191,346530455866,...
1,1,13,247,5813,162624,5417683,217374894,10551425445,618507018238,...
1,1,14,287,7268,217855,7740500,330130230,17007128087,1057156741967,...
1,1,15,330,8947,286014,10791726,487184328,26523926691,1741018836674,...
1,1,16,376,10866,368966,14728894,701255202,40200085065,2776362938533,..
1,1,17,425,13041,468701,19730521,987570893,59420653233,4304220653087,..

Crossrefs

Rows: A124551, A124562, A124563, A124564, A124565, A124566; diagonals: A124567, A124568, A124569; A124561 (antidiagonal sums); variants: A124550, A124460, A124530, A124540.

Programs

  • PARI
    {A124550(n,k)=if(k==0,1,if(n==0,0,if(k==1,n,if(n<=k, Vec(( 1+x*Ser( vector(k,j,sum(i=0,j-1,A124550(n+i*n,j-1-i)) ) ))^n)[k+1], Vec(subst(Ser(concat(concat(0, Vec(subst(Ser(vector(k+1,j,A124550(j-1,k))),x,x/(1+x))/(1+x))),vector(n-k+1)) ),x,x/(1-x))/(1-x +x*O(x^(n))))[n]))))} /* Determined Elements from A124550: */ {T(n,k)=if(n==0|k==0,1,Vec((Ser(vector(k+1,j,A124550(n,j-1)))+x*O(x^k))^(1/n))[k+1])}

Formula

Let F_n(y) be the g.f. of row n in table A124550, then F_n(y) = R_n(y)^n and thus R_n(y) = Sum_{k>=0} y^k * F_{n*k}(y) for n>=0, where R_n(y) is the g.f. of row n in this table.

A124540 Rectangular table, read by antidiagonals, such that the g.f. of row n, R_n(y), satisfies: R_n(y) = [ Sum_{k>=0} y^k * R_k(y)^n ]^n for n>=0, with R_0(y) = 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 5, 0, 1, 4, 15, 26, 16, 0, 1, 5, 26, 73, 107, 62, 0, 1, 6, 40, 156, 369, 486, 274, 0, 1, 7, 57, 285, 939, 1959, 2398, 1332, 0, 1, 8, 77, 470, 1995, 5764, 10912, 12668, 6978, 0, 1, 9, 100, 721, 3756, 13976, 36248, 63543, 70863, 38873, 0
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2006

Keywords

Comments

Antidiagonal sums equal row 1 (A124531).

Examples

			Row g.f.s R_n(y) simultaneously satisfy:
R_n(y) = [1 + y*R_1(y)^n + y^2*R_2(y)^n + y^3*R_3(y)^n +...]^n
more explicitly:
R_0 = [1 + y + y^2 + y^3 + y^4 + ...]^0 = 1;
R_1 = [1 + y*(R_1)^1 + y^2*(R_2)^1 + y^3*(R_3)^1 + y^4*(R_4)^1 +...]^1;
R_2 = [1 + y*(R_1)^2 + y^2*(R_2)^2 + y^3*(R_3)^2 + y^4*(R_4)^2 +...]^2;
R_3 = [1 + y*(R_1)^3 + y^2*(R_2)^3 + y^3*(R_3)^3 + y^4*(R_4)^3 +...]^3;
R_4 = [1 + y*(R_1)^4 + y^2*(R_2)^4 + y^3*(R_3)^4 + y^4*(R_4)^4 +...]^4;
etc., for all rows.
Table begins:
1,0,0,0,0,0,0,0,0,0,0,...
1,1,2,5,16,62,274,1332,6978,38873,228090,...
1,2,7,26,107,486,2398,12668,70863,416304,2552490,...
1,3,15,73,369,1959,10912,63543,385341,2424988,15788469,...
1,4,26,156,939,5764,36248,233900,1549193,10529052,73390856,...
1,5,40,285,1995,13976,98665,704810,5107950,37619020,281850156,...
1,6,57,470,3756,29658,233241,1836912,14543877,116087596,936035298,...
1,7,77,721,6482,57057,495922,4282895,36922550,318834341,2765000007,...
1,8,100,1048,10474,101800,970628,9140344,85445683,795971176,7410928800,...
1,9,126,1461,16074,171090,1777416,18151272,183201255,1834958107,...
1,10,155,1970,23665,273902,3081700,33954660,368443380,3954149640,...
1,11,187,2585,33671,421179,5104528,60398327,701775756,8042277034,...
1,12,222,3316,46557,626028,8133916,102916452,1275653922,15559229828,...

Crossrefs

Rows: A124531, A124542, A124543, A124544, A124545, A124546; diagonals: A124547, A124548, A124549; related tables: A124530, A124550, A124460.

Programs

  • PARI
    T(n,k)=local(m=max(n,k),R);R=vector(m+1,r,vector(m+1,c,if(r==1 || c<=2,1,r^(c-2)))); for(i=0,m, for(r=0,m, R[r+1]=Vec(sum(c=0,m, x^c*Ser(R[c+1])^(r*c)+O(x^(m+1)))))); Vec(Ser(R[n+1])^n+O(x^(k+1)))[k+1]

Formula

Let S_n(y) be the g.f. of row n in table A124530, then R_n(y) = S_n(y)^n and so S_n(y) = Sum_{k>=0} y^k * R_k(y)^n for n>=0, where R_n(y) is the g.f. of row n in this table.
Previous Showing 11-17 of 17 results.

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