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A193196 G.f.: A(x) = Sum_{n>=0} x^n / Product_{k=1..n} (1 - k*x^k). 7
1, 1, 2, 3, 6, 9, 19, 29, 57, 94, 172, 280, 519, 833, 1472, 2433, 4185, 6800, 11666, 18816, 31686, 51340, 84929, 136561, 225476, 359746, 586133, 936243, 1511650, 2397400, 3856698, 6084186, 9711492, 15299490, 24247456, 38016261, 60079125, 93752706, 147284928 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of rooted ordered trees with n non-root nodes such that both successive branch heights and the lengths of the branches are weakly increasing; see example. - Joerg Arndt, Aug 27 2014

LINKS

Robert Israel, Table of n, a(n) for n = 0..5627

Joerg Arndt, trees as in comment for 1<=n<=7

FORMULA

G.f.: G(0) - 1 where G(k) =  1 + (1-x)/(1-x^k*k)/(1-x/(x+(1-x)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 22 2013

a(n) = sum( prod(j=2..m, min(C[j-1], C[j]))) where the sum is over all partitions C[1..m] (m parts) of n, see example. - Joerg Arndt, Sep 03 2014

From Vaclav Kotesovec, Jun 18 2019: (Start)

a(n) ~ c * 3^(n/3), where

c = 9390.8440644933535486959046639452060731482141... if mod(n,3)=0

c = 9390.7389359914729419715573277079935321683397... if mod(n,3)=1

c = 9390.7321933046037554603013237581369727858708... if mod(n,3)=2

(End)

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 9*x^5 + 19*x^6 + 29*x^7 +...

where:

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

From Joerg Arndt, Aug 27 2014: (Start)

The a(4) = 5 trees described in the comment are:

:

:     1:

:    [ 1 1 1 1 ]  <--= branch lengths

:    [ 0 0 0 0 ]  <--= branch heights

:

:  O--o

:  .--o

:  .--o

:  .--o

:

:

:     2:

:    [ 1 1 2 ]

:    [ 0 0 0 ]

:

:  O--o

:  .--o

:  .--o--o

:

:

:     3:

:    [ 1 3 ]

:    [ 0 0 ]

:

:  O--o

:  .--o--o--o

:

:

:     4:

:    [ 2 2 ]

:    [ 0 0 ]

:

:  O--o--o

:  .--o--o

:

:

:     5:

:    [ 2 2 ]

:    [ 0 1 ]

:

:  O--o--o

:     .--o--o

:

:

:     6:

:    [ 4 ]

:    [ 0 ]

:

:  O--o--o--o--o

:

See the Arndt link for all examples for 1 <= n <= 7.

(End)

a(6) = 19 because the 11 partitions of 6 with the products as in the comment are

01:  [ 1 1 1 1 1 1 ]   1*1*1*1*1 = 1

02:  [ 1 1 1 1 2 ]     1*1*1*1   = 1

03:  [ 1 1 1 3 ]       1*1*1     = 1

04:  [ 1 1 2 2 ]       1*1*2     = 2

05:  [ 1 1 4 ]         1*1       = 1

06:  [ 1 2 3 ]         1*2       = 1

07:  [ 1 5 ]           1         = 1

08:  [ 2 2 2 ]         2*2       = 4

09:  [ 2 4 ]           2         = 2

10:  [ 3 3 ]           3         = 3

11:  [ 6 ]         (empty prod.) = 1

and the sum of the products is 19. - Joerg Arndt, Sep 03 2014

MAPLE

N:= 100: # to get all terms up to a(N)

gN:= add(x^n/mul(1-k*x^k, k=1..n), n=0..N):

S:= series(gN, x, N+1):

seq(coeff(S, x, n), n=0..N); # Robert Israel, Aug 28 2014

PROG

(PARI) {a(n)=my(A=1); polcoeff(sum(m=0, n, x^m/prod(k=1, m, 1-k*x^k +x*O(x^n))), n)}

CROSSREFS

Sequence in context: A161704 A011962 A060172 * A319755 A309807 A003243

Adjacent sequences:  A193193 A193194 A193195 * A193197 A193198 A193199

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 17 2011

STATUS

approved

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Last modified October 16 01:52 EDT 2021. Contains 348034 sequences. (Running on oeis4.)