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A217204 Triangle read by rows, related to Bell numbers A000110. 0

%I

%S 1,2,1,5,6,1,15,22,9,2,52,94,63,26,5,203,460,416,244,101,16,877,2532,

%T 2741,2124,1361,384,61,4140,15420,18425,18536,15602,6092,2153,272,

%U 21147,102620,127603,166440,165786,83436,46959,10384,1385,115975,739512,914508,1550864,1700220,1082712,823256,247776,74841,7936

%N Triangle read by rows, related to Bell numbers A000110.

%C See Ma (2012) for precise definition (cf. On combinations of polynomials and Euler numbers).

%H 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 Triangle begins:

%e 1;

%e 2, 1;

%e 5, 6, 1;

%e 15, 22, 9, 2;

%e 52, 94, 63, 26, 5;

%e 203, 460, 416, 244, 101, 16;

%e ...

%t 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 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 M[n_] := P[n] /. {x -> 1, y -> 1};

%t Mbar[n_] := V[n] /. {x -> 1, y -> 1};

%t R[1]=1; R[2] = 2+q; R[n_] := (M[n] /. q -> q^2) + q (Mbar[n] /. q -> q^2);

%t Table[CoefficientList[R[n], q], {n, 1, 10}] // Flatten (* _Jean-Fran├žois Alcover_, Sep 25 2018 *)

%o (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 First column is A000110.

%K nonn,tabl

%O 1,2

%A _N. J. A. Sloane_, Sep 27 2012

%E Example and tabf keyword corrected, and extended by _Michel Marcus_, Feb 11 2013

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Last modified April 23 08:53 EDT 2021. Contains 343204 sequences. (Running on oeis4.)