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A026820 Euler's table: triangular array T read by rows, where T(n,k) = number of partitions in which every part is <=k for 1<=k<=n. Also number of partitions of n into at most k parts. 32
1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 5, 6, 7, 1, 4, 7, 9, 10, 11, 1, 4, 8, 11, 13, 14, 15, 1, 5, 10, 15, 18, 20, 21, 22, 1, 5, 12, 18, 23, 26, 28, 29, 30, 1, 6, 14, 23, 30, 35, 38, 40, 41, 42, 1, 6, 16, 27, 37, 44, 49, 52, 54, 55, 56, 1, 7, 19, 34, 47, 58 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

T(T(n,n),n) = A134737(n). - Reinhard Zumkeller, Nov 07 2007

From Reinhard Zumkeller, Jan 21 2010: (Start)

Row sums give A058397; central terms give A171985;

T(n,1) = A000012(n);

T(n,2) = A008619(n) for n>1;

T(n,3) = A001399(n) for n>2;

T(n,4) = A001400(n) for n>3;

T(n,5) = A001401(n) for n>4;

T(n,6) = A001402(n) for n>5;

T(n,7) = A008636(n) for n>6;

T(n,8) = A008637(n) for n>7;

T(n,9) = A008638(n) for n>8;

T(n,10) = A008639(n) for n>9;

T(n,11) = A008640(n) for n>10;

T(n,12) = A008641(n) for n>11;

T(n,n-2) = A007042(n-1) for n>2;

T(n,n-1) = A000065(n) for n>1;

T(n,n) = A000041(n). (End)

T(A000217(n),n) = A173519(n). - Reinhard Zumkeller, Feb 20 2010

Mirror is A058400. - Omar E. Pol, Apr 21 2012

REFERENCES

G. Chrystal, Algebra, Vol. II, p. 558.

L. Euler, Introductio in Analysin Infinitorum, Book I, chapter XVI.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, p. 493.

LINKS

Alois P. Heinz, Robert G. Wilson v, Rows n = 1..141, flattened

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 831. [scanned copy]

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]

OEIS Wiki, Sorting numbers

R. Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Masterarbeit, Univ. Wien, 2013.

Sergei Viznyuk, C-Program

Sergei Viznyuk, Local copy of C-Program

Eric Weisstein's World of Mathematics, Partition Function q.

Index entries for sequences related to partitions - Reinhard Zumkeller, Jan 21 2010

FORMULA

T(n,k) = T(n,k-1) + T(n-k,k). - Thomas Dybdahl Ahle, Jun 13 2011

T(n,k) = Sum_{i=1..floor(n/2)} T(n-i,i) + Sum_{j=1+floor(n/2)..k} A000041(n-j). - Bob Selcoe, Aug 22 2014

O.g.f.: Product_{i>=0} 1/(1-y*x^i). - Geoffrey Critzer, Mar 11 2012

T(n,k) = A008284(n+k,k). - Álvar Ibeas, Jan 06 2015

EXAMPLE

Triangle starts:

1;

1, 2;

1, 2,  3;

1, 3,  4,  5;

1, 3,  5,  6,  7;

1, 4,  7,  9, 10, 11;

1, 4,  8, 11, 13, 14, 15;

1, 5, 10, 15, 18, 20, 21, 22;

1, 5, 12, 18, 23, 26, 28, 29, 30;

1, 6, 14, 23, 30, 35, 38, 40, 41, 42;

1, 6, 16, 27, 37, 44, 49, 52, 54, 55, 56;

...

MAPLE

T:= proc(n, k) option remember;

      `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))

    end:

seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Apr 21 2012

MATHEMATICA

t[n_, k_] := Length@ IntegerPartitions[n, k]; Table[ t[n, k], {n, 12}, {k, n}] // Flatten

T[n_, k_] := T[n, k] = If[n==0 || k==1, 1, T[n, k-1] + If[k>n, 0, T[n-k, k]]]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 22 2015, after Alois P. Heinz *)

PROG

(Haskell)

import Data.List (inits)

a026820 n k = a026820_tabl !! (n-1) !! (k-1)

a026820_row n = a026820_tabl !! (n-1)

a026820_tabl = zipWith

   (\x -> map (p x) . tail . inits) [1..] $ tail $ inits [1..] where

   p 0 _ = 1

   p _ [] = 0

   p m ks'@(k:ks) = if m < k then 0 else p (m - k) ks' + p m ks

-- Reinhard Zumkeller, Dec 18 2013

CROSSREFS

Partial sums of rows of A008284.

Cf. A026840. - Reinhard Zumkeller, Jan 21 2010

Sequence in context: A109974 A213008 A215520 * A091438 A011794 A221640

Adjacent sequences:  A026817 A026818 A026819 * A026821 A026822 A026823

KEYWORD

nonn,tabl,easy,nice

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified April 28 00:46 EDT 2017. Contains 285556 sequences.