A038321 → basis A007318 → θ+1

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Triangle begins:
  0:            324,             504,              147;
  1:           5832,           13608,             7938,             1372;
  2:         104976,          326592,           285768,            98784,            12005;
  3:        1889568,         7348320,          8573040,          4445280,          1080450,           100842;
  4:       34012224,       158723712,        231472080,        160030080,         58344300,         10890936,          823543;
  5:      612220032,      3333197952,       5833096416,       5040947520,       2450460600,        686128968,       103766418,         6588344;
  6:    11019960576,     68568643584,     139994313984,     145179288576,      88216581600,      32934190464,      7471182096,       948721536,       51883209;
  7:   198359290368,   1388515032576,    3239868409344,    3919840791552,    2858217243840,    1333834713792,    403443833184,     76846444416,     8405079858,     403536070;
  8:  3570467226624,  27770300651520,   72897039210240,  100795906068480,   85746517315200,   48018049696512,  18154972493280,   4610786664960,   756457187220,   72636492600,   3107227739;
  9: 64268410079232, 549851952900096, 1603734862625280, 2494698675194880, 2425401489772800, 1584595639984896, 718936910733888, 228233939915520, 49926174356520, 7191012767400, 615231092322, 23727920916;

The row polynomials satisfy (18+(7*n+14)*x)*P(n,x) = (324+(126*n+504)*x+(49*n+147)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 504*x + 147*x^2.

P[0, x_] := 324 + 504*x + 147*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((324 + (126*n + 504)*x + (49*n + 147)*x^2)*P[n - 1, x])/(18 + (7*n + 14)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 324 + 504*x + 147*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((324 + (126*n + 504)*x + (49*n + 147)*x^2)*v[i - 1])/(18 + (7*n + 14)*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)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.