A124564 -id:A124564 - cp's OEIS frontend

A124554 Row 4 of table A124550; also equals the self-convolution 4th power of A124564, which is row 4 of table A124560.

1, 4, 26, 204, 1899, 21180, 287568, 4802716, 99084889, 2531896840, 80346294380, 3173108251044, 156222183969181, 9603287701405136, 738066706464107408, 71003673625149131020, 8559188942710590217013, 1294012524894298022238700

Offset: 0

Paul D. Hanna, Nov 07 2006

nonn

In table A124550, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = [ Sum_{k>=0} y^k * R_{n*k}(y) ]^n for n>=0.

Cf. A124550; other rows: A124551, A124552, A124553, A124555, A124556.

G.f.: A(x) = [ Sum_{k>=0} y^k * R_{4k}(y) ]^4, where R_n(x) is the g.f. of row n in table A124550.

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_{nk}(y)]^(nk) 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

Paul D. Hanna, Nov 07 2006

			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,..

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

{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])}

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.

A124562 Row 2 of table A124560; also, the self-convolution square equals A124552, which is row 2 of table A124550.

Original entry on oeis.org

1, 1, 3, 12, 63, 429, 3716, 40272, 541655, 9022405, 186233087, 4771577072, 152050410552, 6037181967007, 299176055730253, 18530408350215038, 1436276193993882859, 139462485162718679221, 16980510121067921518710

Offset: 0

Paul D. Hanna, Nov 07 2006

nonn

In table A124560, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k * [R_{n*k}(y)]^(n*k) for n>=0.

Cf. A124560 (table); other rows: A124551, A124563, A124564, A124565, A124566.

G.f.: A(x) = Sum_{k>=0} y^k * [R_{2k}(y)]^(2k), where R_n(x) is the g.f. of row n in table A124560.

A124563 Row 3 of table A124560; also, the self-convolution cube equals A124553, which is row 3 of table A124550.

Original entry on oeis.org

1, 1, 4, 22, 158, 1455, 16918, 244644, 4361883, 95746603, 2592416878, 86825876398, 3607980811184, 186517056299848, 12022039151786966, 967930783674722085, 97490785254375447744, 12299113367136495834881, 1945525636812103772634604

Offset: 0

Paul D. Hanna, Nov 07 2006

nonn

In table A124560, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k * [R_{n*k}(y)]^(n*k) for n>=0.

Cf. A124560 (table); other rows: A124551, A124562, A124564, A124565, A124566.

G.f.: A(x) = Sum_{k>=0} y^k * [R_{3k}(y)]^(3k), where R_n(x) is the g.f. of row n in table A124560.

A124565 Row 5 of table A124560; also, the self-convolution 5th power equals A124555, which is row 5 of table A124550.

Original entry on oeis.org

1, 1, 6, 51, 556, 7581, 128532, 2689248, 68880819, 2155007000, 82603481941, 3896490943878, 227153148813546, 16429403864272555, 1478934508425795630, 166091860417795409081, 23316582876166010185959

Offset: 0

Paul D. Hanna, Nov 07 2006

nonn

In table A124560, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k * [R_{n*k}(y)]^(n*k) for n>=0.

Cf. A124560 (table); other rows: A124551, A124562, A124563, A124564, A124566.

G.f.: A(x) = Sum_{k>=0} y^k * [R_{5k}(y)]^(5k), where R_n(x) is the g.f. of row n in table A124560.

A124566 Row 6 of table A124560; also, the self-convolution 6th power equals A124556, which is row 6 of table A124550.

Original entry on oeis.org

1, 1, 7, 70, 891, 14036, 272914, 6525900, 190604859, 6781448755, 294798563020, 15737487680990, 1036588563202854, 84606134756948277, 8587502188940359207, 1086820294948914428468, 171866738763640156327659

Offset: 0

Paul D. Hanna, Nov 07 2006

nonn

In table A124560, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k * [R_{n*k}(y)]^(n*k) for n>=0.

Cf. A124560 (table); other rows: A124551, A124562, A124563, A124564, A124565.

G.f.: A(x) = Sum_{k>=0} y^k * [R_{6k}(y)]^(6k), where R_n(x) is the g.f. of row n in table A124560.

cp's OEIS Frontend

A124554 Row 4 of table A124550; also equals the self-convolution 4th power of A124564, which is row 4 of table A124560.

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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).

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PARI

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A124562 Row 2 of table A124560; also, the self-convolution square equals A124552, which is row 2 of table A124550.

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A124563 Row 3 of table A124560; also, the self-convolution cube equals A124553, which is row 3 of table A124550.

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A124565 Row 5 of table A124560; also, the self-convolution 5th power equals A124555, which is row 5 of table A124550.

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A124566 Row 6 of table A124560; also, the self-convolution 6th power equals A124556, which is row 6 of table A124550.

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Formula

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_{nk}(y)]^(nk) for n>=0, with R_0(y)=1/(1-y).