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A344911 Concatenated Bessel-scaled Pascal triangles. Irregular triangle read by rows, T(n,k) with n >= 0 and 0 <= k <= (2*n*(n + 4) - 1 + (-1)^n)/8.

%I A344911 #18 Jun 06 2021 15:46:21
%S A344911 1,1,1,1,2,1,1,1,3,3,1,3,3,1,4,6,4,1,6,12,6,3,1,5,10,10,5,1,10,30,30,
%T A344911 10,15,15,1,6,15,20,15,6,1,15,60,90,60,15,45,90,45,15,1,7,21,35,35,21,
%U A344911 7,1,21,105,210,210,105,21,105,315,315,105,105,105
%N A344911 Concatenated Bessel-scaled Pascal triangles. Irregular triangle read by rows, T(n,k) with n >= 0 and 0 <= k <= (2*n*(n + 4) - 1 + (-1)^n)/8.
%C A344911 Let p(n) = Sum_{k=0..n/2} Sum_{j=0..n-2*k} (n!/(2^k*k!*j!*(n-2*k-j)!))*x^j*y^(n-2*k-j). Row n of the triangle is defined as the coefficient list of the polynomials p(n), where the monomials are in degree-lexicographic order.
%C A344911 One observes: The triangle of the coefficients appears as a series of concatenated subtriangles. The first one is Pascal's triangle A007318. Appending the rows of triangle A094305 on the right side starts in row 2. In row 4, the next triangle is appended, which is A344565. This scheme goes on indefinitely.
%C A344911 This can be formalized as follows: Let C(n) denote row n of the binomial triangle, set c.C(n) = Seq_{j=0..n} c*binomial(n, j), and let B(n, k) denote the Bessel numbers A100861(n, k). Then T(n) = Seq_{k=0..n/2} B(n, k).C(n-2*k). Since B(n, k) = binomial(n, 2*k)*(2*k - 1)!! it follows that: T(n) = Seq_{k=0..n/2} Seq_{j=0..n-2*k} binomial(n, 2*k)*binomial(n-2*k, j)*(2*k-1)!!. This expression equals the coefficient list of p(n) since the monomials are in degree-lexicographic order.
%C A344911 The polynomials are also the unsigned, probabilist's Hermite polynomials H_n(x+y)
%C A344911 which are discussed in A344678. The coefficients are listed there in a different order which do not reveal the structure described above.
%F A344911 The bivariate e.g.f. exp(t^2/2)*exp(t*(x + y)) = Sum_{n>=0} H_n(x + y)*t^n/n!, where H_n(x) are the unsigned, modified Hermite polynomials A099174, is given by _Tom Copeland_ in A344678.
%e A344911 The triangle begins:
%e A344911 [0] [ 1 ]
%e A344911 [1] [ 1, 1 ]
%e A344911 [2] [ 1, 2,  1 ][ 1 ]
%e A344911 [3] [ 1, 3,  3,   1 ][ 3,   3 ]
%e A344911 [4] [ 1, 4,  6,   4,   1 ][ 6,   12,    6 ][ 3 ]
%e A344911 [5] [ 1, 5, 10,  10,   5,   1 ][ 10,   30,  30, 10 ][ 15, 15 ]
%e A344911 [6] [ 1, 6, 15,  20,  15,   6,    1 ][ 15,  60, 90,   60, 15 ][ 45, 90, 45][ 15 ]
%e A344911 .
%e A344911 With the notations in the comment row 7 concatenates:
%e A344911 B(7, 0).C(7) =   1.[1, 7, 21, 35, 35, 21, 7, 1] = [1, 7, 21, 35, 35, 21, 7, 1],
%e A344911 B(7, 1).C(5) =  21.[1, 5, 10, 10, 5, 1]         = [21, 105, 210, 210, 105, 21],
%e A344911 B(7, 2).C(3) = 105.[1, 3, 3, 1]                 = [105, 315, 315, 105],
%e A344911 B(7, 3).C(1) = 105.[1, 1]                       = [105, 105].
%e A344911 .
%e A344911 p_6(x,y) = x^6 + 6*x^5*y + 15*x^4*y^2 + 20*x^3*y^3 + 15*x^2*y^4 + 6*x*y^5 + y^6 +
%e A344911 15*x^4 + 60*x^3*y + 90*x^2*y^2 + 60*x*y^3 + 15*y^4 + 45*x^2 + 90*x*y + 45*y^2 + 15.
%p A344911 P := n -> add(add(n!/(2^k*k!*j!*(n-2*k-j)!)*y^(n-2*k-j)*x^j, j=0..n-2*k), k=0..n/2):
%p A344911 seq(seq(subs(x = 1, y = 1, m), m = [op(P(n))]), n = 0..7);
%p A344911 # Alternatively, without polynomials:
%p A344911 B := (n, k) -> binomial(n, 2*k)*doublefactorial(2*k-1):
%p A344911 C := n -> seq(binomial(n, j), j=0..n):
%p A344911 seq(seq(B(n, k)*C(n-2*k), k = 0..n/2), n = 0..7);
%p A344911 # Based on the e.g.f. of the polynomials:
%p A344911 T := proc(numofrows) local gf, ser, n, m;
%p A344911 gf := exp(t^2/2)*exp(t*(x + y)); ser := series(gf, t, numofrows+1);
%p A344911 for n from 0 to numofrows do [op(sort(n!*expand(coeff(ser, t, n))))];
%p A344911 print(seq(subs(x=1, y=1, m), m = %)) od end: T(7);
%t A344911 P[n_] := Sum[ Sum[n! / (2^k k! j! (n - 2k - j)!) y^(n - 2k - j) x^j, {j, 0, n-2k}], {k, 0, n/2}];
%t A344911 DegLexList[p_] := MonomialList[p, {x, y}, "DegreeLexicographic"] /. x->1 /. y->1;
%t A344911 Table[DegLexList[P[n]], {n, 0, 7}] // Flatten
%Y A344911 Cf. A005425 (row sums), A100861 (scaling factors).
%Y A344911 Cf. A007318, A094305, A099174, A344565, A344678.
%K A344911 nonn,tabf
%O A344911 0,5
%A A344911 _Peter Luschny_, Jun 03 2021