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 A334774 Triangle read by rows: T(n,k) is the number of permutations of 2 indistinguishable copies of 1..n with exactly k local maxima. 26
 1, 3, 3, 9, 57, 24, 27, 705, 1449, 339, 81, 7617, 48615, 49695, 7392, 243, 78357, 1290234, 3650706, 2234643, 230217, 729, 791589, 30630618, 197457468, 314306943, 128203119, 9689934, 2187, 7944321, 686779323, 9080961729, 30829608729, 31435152267, 9159564513, 529634931 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also the number of permutations of 2 indistinguishable copies of 1..n with exactly k-1 peaks. A peak is an interior maximum. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50) FORMULA T(n,k) = F(2,n,k-1,0) where F(m,n,p,q) = Sum_{i=0..p} Sum_{j=0..min(m-i, q)} F(m, n-1, p-i, q-j+i) * binomial(m+2*(q-j)+1, 2*q+i-j+1) * binomial(q-j+i, i) * binomial(q+1, j) for n > 1 with F(m,1,0,q) = binomial(m-1, q), F(m,1,p,q) = 0 for p > 0. A334776(n) = Sum_{k=1..n} (k-1)*T(n,k). A334777(n) = Sum_{k=1..n} k*T(n,k). EXAMPLE Triangle begins:     1;     3,      3;     9,     57,       24;    27,    705,     1449,       339;    81,   7617,    48615,     49695,      7392;   243,  78357,  1290234,   3650706,   2234643,    230217;   729, 791589, 30630618, 197457468, 314306943, 128203119, 9689934;   ... The T(2,1) = 3 permutations of 1122 with 1 local maxima are 1122, 1221, 2211. The T(2,2) = 3 permutations of 1122 with 2 local maxima are 1212, 2112, 2121. The T(2,1) = 3 permutations of 1122 with 0 peaks are 2211, 2112, 1122. The T(2,2) = 3 permutations of 1122 with 1 peak are 2121, 1221, 1212. PROG (PARI) PeaksBySig(sig, D)={   my(F(lev, p, q) = my(key=[lev, p, q], z); if(!mapisdefined(FC, key, &z),     my(m=sig[lev]); z = if(lev==1, if(p==0, binomial(m-1, q), 0), sum(i=0, p, sum(j=0, min(m-i, q), self()(lev-1, p-i, q-j+i) * binomial(m+2*(q-j)+1, 2*q+i-j+1) * binomial(q-j+i, i) * binomial(q+1, j) )));     mapput(FC, key, z)); z);   local(FC=Map());   vector(#D, i, F(#sig, D[i], 0)); } Row(n)={ PeaksBySig(vector(n, i, 2), [0..n-1]) } { for(n=1, 8, print(Row(n))) } CROSSREFS Columns k=1..6 are A000244(n-1), 3*A152494, 3*A152495, 3*A152496, 3*A152497, 3*A152498. Row sums are A000680. Main diagonal is A334775. The version for permutations of 1..n is A008303(n,k-1). Cf. A154283, A334773, A334776, A334777, A334778. Sequence in context: A257623 A257625 A216147 * A257627 A115564 A122961 Adjacent sequences:  A334771 A334772 A334773 * A334775 A334776 A334777 KEYWORD nonn,tabl AUTHOR Andrew Howroyd, May 11 2020 STATUS approved

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Last modified December 7 22:50 EST 2021. Contains 349590 sequences. (Running on oeis4.)