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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. 33
 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. FORMULA 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 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. 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,nice,changed AUTHOR STATUS approved

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Last modified March 22 14:47 EDT 2018. Contains 301071 sequences. (Running on oeis4.)