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A101509 Binomial transform of tau(n) (see A000005).

%I A101509 #47 Dec 23 2024 14:53:42
%S A101509 1,3,7,16,35,75,159,334,696,1442,2976,6123,12562,25706,52492,107014,
%T A101509 217877,443061,899957,1826078,3701783,7498261,15178255,30706320,
%U A101509 62085915,125465715,253415981,511608490,1032427637,2082680887,4199956101,8467124805,17064784905,34382825363,69256687719,139465867773
%N A101509 Binomial transform of tau(n) (see A000005).
%C A101509 Row sums of A101508.
%C A101509 Also: Number of matrices with positive integer coefficients such that the sum of all entries equals n+1, cf. link "Partitions and A101509". - _M. F. Hasler_, Jan 14 2009
%H A101509 M. F. Hasler, <a href="/A101509/b101509.txt">Table of n, a(n) for n = 0..500</a>
%H A101509 L. Manor, M. F. Hasler, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2009-January/000522.html">Partitions and A101509. SeqFan list, Jan 14 2009</a>
%H A101509 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A101509 a(n) = Sum_{k=0..n, Sum_{i=0..n, if(mod(i+1, k+1)=0, binomial(n, i), 0)}}.
%F A101509 G.f.: 1/x * Sum_{n>=1} z^n/(1-z^n) (Lambert series) where z=x/(1-x). - _Joerg Arndt_, Jan 30 2011
%F A101509 a(n) ~ 2^n * (log(n/2) + 2*gamma), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Mar 07 2020
%e A101509 From _Gus Wiseman_, Jan 16 2019: (Start)
%e A101509 The a(3) = 16 ways to arrange the parts of an integer partition of 4 into a matrix:
%e A101509   [4] [1 3] [3 1] [2 2] [1 1 2] [1 2 1] [2 1 1] [1 1 1 1]
%e A101509 .
%e A101509   [1] [3] [2] [1 1]
%e A101509   [3] [1] [2] [1 1]
%e A101509 .
%e A101509   [1] [1] [2]
%e A101509   [1] [2] [1]
%e A101509   [2] [1] [1]
%e A101509 .
%e A101509   [1]
%e A101509   [1]
%e A101509   [1]
%e A101509   [1]
%e A101509 (End)
%p A101509 bintr:= proc(p) proc(n) add(p(k) *binomial(n, k), k=0..n) end end:
%p A101509 a:= bintr(n-> numtheory[tau](n+1)):
%p A101509 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jan 30 2011
%t A101509 a[n_] := Sum[DivisorSigma[0, k+1]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Feb 18 2017 *)
%o A101509 (PARI) A101509(n) = sum( k=0,n, numdiv(k+1)*binomial(n,k)) \\ _M. F. Hasler_, Jan 14 2009
%Y A101509 Cf. A000005 (tau), A101508, A160399.
%Y A101509 Cf. A000219, A047966, A053529, A319066, A323300, A323301, A323307, A323429.
%K A101509 easy,nonn
%O A101509 0,2
%A A101509 _Paul Barry_, Dec 05 2004