A114102 -id:A114102 - cp's OEIS frontend

A156144 Number of partitions of n into parts having in decimal representation the same digital root as n has.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 3, 2, 3, 1, 1, 2, 1, 1, 3, 5, 2, 5, 1, 1, 2, 1, 1, 5, 8, 4, 8, 2, 1, 4, 1, 1, 7, 13, 5, 13, 2, 2, 5, 1, 1, 11, 20, 9, 19, 3, 2, 9, 1, 1, 15, 31, 12, 29, 4, 3, 11, 2, 1, 22, 46, 20, 42, 7, 4, 18, 2, 2, 30, 68, 27, 61, 9, 6, 23, 3, 2, 42, 98, 42, 85

Offset: 1

Reinhard Zumkeller, Feb 05 2009

a(n) <= a(n+9); Max{n: a(n)=1} = 71;
A156145 and A017173 give record values and where they occur: a(A017173(n-1))=A156145(n);
a(A017173(n)) = A116371(A017173(n)).

			a(19) = #{19, 10+1+1+1+1+1+1+1+1+1, 19x1} = 3;
a(20) = #{20, 2+2+2+2+2+2+2+2+2+2} = 2;
a(21) = #{21, 3+3+3+3+3+3+3, 12+3+3+3} = 3;
a(22) = #{22} = 1;

R. Zumkeller, Table of n, a(n) for n = 1..500

A010888, A114102.

a156144 n = p [x | x <- [1..n], a010888 x == a010888 n] n where
   p _  0 = 1
   p [] _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Feb 04 2014

A224106 Numerators of poly-Cauchy numbers of the second kind hat c_n^(4).

Original entry on oeis.org

1, -1, 97, -1147, 3472243, -653983, 74118189437, -1058923294571, 777910456216513, -285577840060819, 23240203016832136201, -216925341603548096639, 1222007019804929270080450811

Offset: 0

Takao Komatsu, Mar 31 2013

The poly-Cauchy numbers of the second kind hat c_n^(k) can be expressed in terms of the (unsigned) Stirling numbers of the first kind: hat c_n^(k) = (-1)^n*sum(abs(stirling1(n,m))/(m+1)^k, m=0..n).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Takao Komatsu, Poly-Cauchy numbers, RIMS Kokyuroku 1806 (2012)
Takao Komatsu, Poly-Cauchy numbers with a q parameter, Ramanujan J. 31 (2013), 353-371.
Takao Komatsu, Poly-Cauchy numbers, Kyushu J. Math. 67 (2013), 143-153.
T. Komatsu, V. Laohakosol, and K. Liptai, A generalization of poly-Cauchy numbers and its properties, Abstract and Applied Analysis, Volume 2013, Article ID 179841, 8 pages.
Takao Komatsu and F.-Z. Zhao, The log-convexity of the poly-Cauchy numbers, arXiv preprint arXiv:1603.06725 [math.NT], 2016.

Cf. A002657, A223902, A224105, A114102, A224104, A224105 (denominators).

Table[Numerator[Sum[StirlingS1[n, k] (-1)^k/ (k + 1)^4, {k, 0, n}]], {n, 0,
  25}]

a(n) = numerator(sum(k=0, n,(-1)^k*stirling(n, k, 1)/(k+1)^4)); \\ Michel Marcus, Nov 15 2015

cp's OEIS Frontend

A156144 Number of partitions of n into parts having in decimal representation the same digital root as n has.

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