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A110582 Triangle, read by rows, where the g.f. of diagonal n, D_n(x), is generated from the g.f. of row n-1, R_{n-1}(x), by D_n(x) = R_{n-1}(x)/(1-x)^2 for n>0, with D_0(x) = 1/(1-x)^2. 1
1, 1, 2, 1, 2, 3, 1, 4, 3, 4, 1, 4, 7, 4, 5, 1, 6, 10, 10, 5, 6, 1, 6, 14, 16, 13, 6, 7, 1, 8, 18, 26, 22, 16, 7, 8, 1, 8, 25, 34, 38, 28, 19, 8, 9, 1, 10, 29, 52, 55, 50, 34, 22, 9, 10, 1, 10, 37, 66, 84, 76, 62, 40, 25, 10, 11, 1, 12, 44, 90, 116, 122, 97, 74, 46, 28, 11, 12 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Related to planar partitions.

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

FORMULA

T(n, k) = Sum_{j=0..k} T(n-k-1, j)*(k-j+1) with T(n, n) = n+1.

G.f.: A(x, y) = Sum_{j=0..n} x^j/Product_{i=1..j+1} (1-y*x^i)^2.

EXAMPLE

Triangle begins:

1;

1,2;

1,2,3;

1,4,3,4;

1,4,7,4,5;

1,6,10,10,5,6;

1,6,14,16,13,6,7;

1,8,18,26,22,16,7,8;

1,8,25,34,38,28,19,8,9;

1,10,29,52,55,50,34,22,9,10; ...

Row sums form A006330 (offset 1):

{1,3,6,12,21,38,63,106,170,272,422,653,...},

(planar partitions with only one row and one column).

G.f. of diagonal n, D_n(x), is generated from g.f. of

row n-1, R_{n-1}(x), by D_n(x) = R_{n-1}(x)/(1-x)^2:

D_3(x) = 1 + 4*x + 10*x^2 + 16*x^3 + 22*x^4 + ...

= (1 + 2*x + 3*x^2)/(1-x)^2 = R_2(x)/(1-x)^2;

D_4(x) = 1 + 6*x + 14*x^2 + 26*x^3 + 38*x^4 + ...

= (1+ 4*x+ 3*x^2+ 4*x^3)/(1-x)^2 = R_3(x)/(1-x)^2.

MATHEMATICA

T[n_, k_] := T[n, k] = If[n < k || k < 0, 0, If[k == 0, 1, If[k == n, n + 1, Sum[T[n - k - 1, j]*(k - j + 1), {j, 0, k}]]]]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Sep 01 2017 *)

PROG

(PARI) T(n, k)=if(n<k || k<0, 0, if(k==0, 1, if(k==n, n+1, sum(j=0, k, T(n-k-1, j)*(k-j+1))); ))

(PARI) T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); if(n<k || k<0, 0, polcoeff(polcoeff(sum(j=0, n, X^j/prod(i=1, j+1, 1-Y*X^i)^2), n, x), k, y))

CROSSREFS

Cf. A006330 (row sums).

Sequence in context: A205682 A217762 A172467 * A162507 A091298 A306641

Adjacent sequences:  A110579 A110580 A110581 * A110583 A110584 A110585

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Jul 29 2005

STATUS

approved

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Last modified April 8 05:55 EDT 2020. Contains 333312 sequences. (Running on oeis4.)