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A217204 Triangle read by rows, related to Bell numbers A000110: A216962 interlaced with A216964.

%I A217204 #32 Jan 13 2025 12:29:45
%S A217204 1,2,1,5,6,1,15,22,9,2,52,94,63,26,5,203,460,416,244,101,16,877,2532,
%T A217204 2741,2124,1361,384,61,4140,15420,18425,18536,15602,6092,2153,272,
%U A217204 21147,102620,127603,166440,165786,83436,46959,10384,1385,115975,739512,914508,1550864,1700220,1082712,823256,247776,74841,7936
%N A217204 Triangle read by rows, related to Bell numbers A000110: A216962 interlaced with A216964.
%C A217204 See Ma (2012) for precise definition (cf. On combinations of polynomials and Euler numbers).
%H A217204 S.-M. Ma, <a href="https://arxiv.org/abs/1203.6264">Enumeration of permutations by number of cyclic peaks and cyclic valleys</a>, arXiv preprint arXiv:1203.6264 [math.CO], 2012.
%e A217204 Triangle begins:
%e A217204     1;
%e A217204     2,   1;
%e A217204     5,   6,   1;
%e A217204    15,  22,   9,   2;
%e A217204    52,  94,  63,  26,   5;
%e A217204   203, 460, 416, 244, 101, 16;
%e A217204   ...
%t A217204 P[1] := x y; P[n_] := P[n] = ((n-1) q + x y) P[n-1] + 2 q (1-q) D[P[n-1], q] + x (1-q) D[P[n-1], x] + (1-y) D[P[n-1], y] // Simplify;
%t A217204 V[1] = x y; V[n_] := V[n] = ((n-1) q + x y) V[n-1] + 2 q (1-q) D[V[n-1], q] + 2 x (1-q) D[V[n-1], x] + (1 - 2 y + q y) D[V[n-1], y] // Simplify;
%t A217204 M[n_] := P[n] /. {x -> 1, y -> 1};
%t A217204 Mbar[n_] := V[n] /. {x -> 1, y -> 1};
%t A217204 R[1]=1; R[2] = 2+q; R[n_] := (M[n] /. q -> q^2) + q (Mbar[n] /. q -> q^2);
%t A217204 Table[CoefficientList[R[n], q], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Sep 25 2018 *)
%o A217204 (PARI) tabl(m) = {Pa = x; Pb = x*y; for (n=1, m, Pa1 = subst(Pa, x, 1); Pb1 = subst(Pb, x, 1); Pb1 = subst(Pb1, y, 1); if (n==1, R = 1, if (n==2, R = 2+q, R = subst(Pa1, q, q^2) + q*subst(Pb1, q, q^2););); for (d=0, poldegree(R, q), print1(polcoeff(R, d, q), ", "); ); print(""); Pa = (n*q+x)*Pa + 2*q*(1-q)*deriv(Pa, q)+ x*(1-q)*deriv(Pa,x); Pb = (n*q+x*y)*Pb + 2*q*(1-q)*deriv(Pb, q)+ 2*x*(1-q)*deriv(Pb,x)+ (1-2*y+q*y)*deriv(Pb,y););} \\ _Michel Marcus_, Feb 11 2013
%Y A217204 First column is A000110.
%Y A217204 Cf. A216962, A216964.
%K A217204 nonn,tabl
%O A217204 1,2
%A A217204 _N. J. A. Sloane_, Sep 27 2012
%E A217204 Example and tabf keyword corrected, and extended by _Michel Marcus_, Feb 11 2013