This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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. 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 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

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