This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A209936 #42 Jan 29 2025 12:12:52 %S A209936 1,2,1,3,6,1,4,12,6,12,1,5,20,20,30,30,20,1,6,30,30,60,15,120,60,20, %T A209936 90,30,1,7,42,42,105,42,210,140,105,105,420,105,140,210,42,1,8,56,56, %U A209936 168,56,336,280,28,336,168,840,280,168,420,840,1120,168,70,560,420,56,1 %N A209936 Triangle of multiplicities of k-th partition of n corresponding to sequence A080577. Multiplicity of a given partition of n into k parts is the number of ways parts can be selected from k distinguishable bins. See the example. %C A209936 Differs from A035206 after position 21. %C A209936 Differs from A210238 after position 21. %C A209936 The n-th row of the triangle, written as a column vector v(n), satisfies K . v(n) = #SSYT(lambda,n) where K is the Kostka matrix of order n, and #SSYT(lambda,n) is the count of semi-standard Young tableaux in n variables of the partitions of n. - _Wouter Meeussen_, Jan 27 2025 %H A209936 Sergei Viznyuk, <a href="http://phystech.com/ftp/s_A209936.c">C Program</a> %e A209936 Triangle begins: %e A209936 1 %e A209936 2, 1 %e A209936 3, 6, 1 %e A209936 4, 12, 6, 12, 1 %e A209936 5, 20, 20, 30, 30, 20, 1 %e A209936 6, 30, 30, 60, 15, 120, 60, 20, 90, 30, 1 %e A209936 7, 42, 42, 105, 42, 210, 140, 105, 105, 420, 105, 140, 210, 42, 1 %e A209936 ... %e A209936 Thus for n=3 (third row) the partitions of n=3 are: %e A209936 3+0+0 0+3+0 0+0+3 (multiplicity=3), %e A209936 2+1+0 2+0+1 1+2+0 1+0+2 0+2+1 0+1+2 (multiplicity=6), %e A209936 1+1+1 (multiplicity=1). %t A209936 Apply[Multinomial,Last/@Tally[#]&/@PadRight[IntegerPartitions[n]],1] (* _Wouter Meeussen_, Jan 26 2025 *) %Y A209936 Cf. A080577, A078760, A035206, A210238. %Y A209936 Row lengths give A000041. %Y A209936 Row sums give A088218. %K A209936 nonn,tabf %O A209936 1,2 %A A209936 _Sergei Viznyuk_, Mar 15 2012