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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. 34
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

REFERENCES

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

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]

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

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

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

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

T(n,k) = Sum_{i=1..min(k,floor(n/2))} T(n-i,i) + Sum_{j=1+floor(n/2)..k} A000041(n-j). - Bob Selcoe, Aug 22 2014 [corrected by Álvar Ibeas, Mar 15 2018]

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

(* Second program: *)

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

(PARI) T(n, k)=my(s); forpart(v=n, s++, , k); s \\ Charles R Greathouse IV, Feb 27 2018

CROSSREFS

Partial sums of rows of A008284, row sums give A058397, central terms give A171985, mirror is A058400.

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

Cf. A008284, A026840, A134737, A173519.

Sequence in context: A109974 A213008 A215520 * A091438 A011794 A221640

Adjacent sequences:  A026817 A026818 A026819 * A026821 A026822 A026823

KEYWORD

nonn,tabl,nice

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified October 16 16:49 EDT 2018. Contains 316269 sequences. (Running on oeis4.)