A035342: The convolution matrix of the double factorial of odd numbers (A001147).

OEIS matches ?

• A049118: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
• A035342: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0:         1;
  1:         3,          1;
  2:        15,          9,         1;
  3:       105,         87,        18,         1;
  4:       945,        975,       285,        30,        1;
  5:     10395,      12645,      4680,       705,       45,       1;
  6:    135135,     187425,     82845,     15960,     1470,      63,      1;
  7:   2027025,    3133935,   1595790,    370125,    43890,    2730,     84,    1;
  8:  34459425,   58437855,  33453945,   8998290,  1291815,  103950,   4662,  108,   1;
  9: 654729075, 1203216525, 760970700, 231416325, 38710035, 3786615, 220500, 7470, 135, 1;

The row polynomials satisfy P(n,x) = (2*n+1+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n + 1 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n + 1 + x)*v[i - 1] + x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Triangle begins:
   1:         1;
   2:         3,          1;
   3:        15,          9,         1;
   4:       105,         87,        18,         1;
   5:       945,        975,       285,        30,        1;
   6:     10395,      12645,      4680,       705,       45,       1;
   7:    135135,     187425,     82845,     15960,     1470,      63,      1;
   8:   2027025,    3133935,   1595790,    370125,    43890,    2730,     84,    1;
   9:  34459425,   58437855,  33453945,   8998290,  1291815,  103950,   4662,  108,   1;
  10: 654729075, 1203216525, 760970700, 231416325, 38710035, 3786615, 220500, 7470, 135, 1;

The row polynomials satisfy P(n,x) = (2*n-1+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n - 1 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2*n - 1 + x)*v[i - 1] + x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-cf2bc165f5c18b21
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-cf2bc165f5c18b21
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-cf2bc165f5c18b21
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-cf2bc165f5c18b21
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 0] to roots in (-inf, 0]: (-inf, 0] -> (-inf, 0] through R_s, s = 2, and U_g
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-9a325246a0b0b95a

Proof chains

2 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map_theorem e-9a325246a0b0b95a root: theorem check

  Index-dependent basis map [Bre89, Thm 2.4.3; MW08, Thm 2] polytool RealRooted basisMap_factorization_of_scalarIndex_recurrence RealRooted rayBasisMap_maps_realRootedIn_iff ?

• theorem_check liu_wang e-cf2bc165f5c18b21 root: theorem check

  Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Used in 2 proof steps

• e-91082a28c5d8ee02 obreschkoff_combination → obreschkoff_combination of The convolution matrix of the double factorial of odd numbers …
• e-91082a28c5d8ee02 obreschkoff_combination → obreschkoff_combination of The convolution matrix of the double factorial of odd numbers …

Related sequences ?

No candidate equalities or related sequences.

References

1. [Bre89] Francesco Brenti. Unimodal, log-concave and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc., 81(413):viii+106, 1989. doi:10.1090/memo/0413
2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741