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A094533 Number of one-element transitions among partitions of the integer n for labeled parts. 4
0, 0, 2, 8, 22, 48, 98, 178, 316, 524, 856, 1334, 2066, 3084, 4578, 6626, 9530, 13434, 18854, 26022, 35764, 48520, 65526, 87550, 116536, 153674, 201906, 263258, 342006, 441366, 567754, 726032, 925588, 1174010, 1484664, 1869072, 2346586, 2934044 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, q-Polygamma Function, q-Pochhammer Symbol

FORMULA

a(n) = Sum_p=1^P(n) Sum_i=1^D(p) Sum_j=i^D(p) 1 [subject to: d(i, p) <= d(j, p) ]; P(n) = number of partitions of n, D(p) = number of digits in partition p, d(i, p) and d(j, p) = digits number i and j in partition p of integer n.

a(n) = Sum_i=1^P(n) p(i, n)^2 - p(i, n), where P(n) is the number of integer partitions of n and p(i, n) is the number of parts of the i-th partition of n.

G.f.: (log(1-x)^2 - log(1-x)*log(x) + psi_x(1)*(2*log(1-x) - log(x) + psi_x(1)) + psi^1_x(1))/((x; x)_inf * log(x)^2), where psi_q(z) is the q-digamma function, psi^1_q(z) is the q-trigamma function, and (a; q)_inf is the q-Pochhammer symbol (the Euler function). To get this g.f., take the derivative (d/da)^2 (x^2/(a; x)_inf) and let a = x. - Vladimir Reshetnikov, Nov 21 2016

EXAMPLE

In the labeled case we have 22 one-element transitions among all partitions of n=4:

[1,1,1,1] -> [1,1,2] arises 6 times (the first 1 added to the second 1 gives 2,

the first 1 added to the third 1 gives 2, the first 1 added to the fourth 1 gives 2, the second 1 added to the third 1 gives 2, the second 1 added to the fourth 1 gives 2, the third 1 added to the fourth 1 gives 2),

[1,1,2] -> [2,2] arises 1 times,

[1,1,2] -> [1,3] arises 2 times,

[2,2] -> [1,3] arises 1 times,

[1,3] -> [4] arises 1 time,

which gives 11 upwards transitions and 22 transitions in total if we include downwards transitions.

n=4: partition number p=1 is [1,1,1,1],

digits d(1,1)=1, d(2,1)=1 contribute 1,

digits d(1,1)=1, d(3,1)=1 contribute 1,

etc...

digits d(3,1)=1, d(4,1)=1 contribute 1,

(in total 6 contributions by [1,1,1,1]);

partition number p=2 is [1,1,2],

digits d(1,2)=1, d(2,2)=1 contribute 1,

digits d(1,2)=1, d(3,2)=2 contribute 1,

digits d(2,2)=1, d(3,2)=2 contribute 1;

partition number p=3 is [2,2],

digits d(1,3)=2, d(2,3)=2 contribute 1;

partition number p=4 is [1,3],

digits d(1,4)=1, d(2,4)=3 contribute 1;

partition number p=5 is [4],

digit d(1,5)=4 contributes 0;

MAPLE

main := proc(n::integer) local a, ndxp, ListOfPartitions, APartition, PartOfAPartition; with(combinat): ListOfPartitions:=partition(n); a:=0; for ndxp from 1 to nops(ListOfPartitions) do APartition := ListOfPartitions[ndxp]; a := a + nops(APartition)^2 - nops(APartition); end do; print("n, a(n):", n, a); end proc;

# second Maple program:

b:= proc(n, i) option remember; local f, g;

      if n=0 then [1, [1]] elif i<1 then [0, [0]]

    else f:= b(n, i-1); g:= `if`(i>n, [0, [0]], b(n-i, i));

         [f[1]+g[1], zip((x, y)-> x+y, f[2], [0, g[2][]], 0)]

      fi

    end:

a:= n-> (l-> add(l[t+1]*t*(t-1), t=1..nops(l)-1))(b(n$2)[2]):

seq(a(n), n=0..50);  # Alois P. Heinz, Apr 05 2012

MATHEMATICA

a[n_] := Block[{p = IntegerPartitions[n], l = PartitionsP[n]}, Sum[ Length[p[[k]]]^2 - Length[p[[k]]], {k, l}]]; Table[ a[n], {n, 0, 37}] (* Robert G. Wilson v, Jul 13 2004, updated by Jean-Fran├žois Alcover, Jan 29 2014 *)

Simplify@Table[SeriesCoefficient[(Log[1 - x]^2 - Log[1 - x] Log[x] + QPolyGamma[1, x] (2 Log[1 - x] - Log[x] + QPolyGamma[1, x]) + QPolyGamma[1, 1, x])/(QPochhammer[x] Log[x]^2), {x, 0, n}], {n, 0, 40}] (* Vladimir Reshetnikov, Nov 21 2016 *)

Simplify@Table[SeriesCoefficient[2 q^2/QPochhammer[q + a, q], {a, 0, 2}, {q, 0, n}], {n, 0, 40}] (* Vladimir Reshetnikov, Nov 22 2016 *)

CROSSREFS

Cf. A093695.

Sequence in context: A284922 A212970 A212683 * A006696 A094939 A006732

Adjacent sequences:  A094530 A094531 A094532 * A094534 A094535 A094536

KEYWORD

nonn

AUTHOR

Thomas Wieder, Jun 05 2004

EXTENSIONS

More terms from Robert G. Wilson v, Jul 13 2004

STATUS

approved

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Last modified April 29 15:47 EDT 2017. Contains 285607 sequences.