A038338 → x ↦ x+1 → θ+1

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Submission text ?

Triangle begins:
  0:              576,              1152,               432;
  1:            13824,             41472,             31104,               6912;
  2:           331776,           1327104,           1492992,             663552,             103680;
  3:          7962624,          39813120,          59719680,           39813120,           12441600,            1492992;
  4:        191102976,        1146617856,        2149908480,         1911029760,          895795200,          214990848,          20901888;
  5:       4586471424,       32105299968,       72236924928,        80263249920,        50164531200,        18059231232,        3511517184,         286654464;
  6:     110075314176,      880602513408,     2311581597696,      3082108796928,      2407897497600,      1155790798848,      337105649664,       55037657088,       3869835264;
  7:    2641807540224,    23776267862016,    71328803586048,    110955916689408,    104021171896320,     62412703137792,    24271606775808,     5944066965504,     835884417024,      51597803520;
  8:   63403380965376,   634033809653760,  2139864107581440,   3804202857922560,   4160846875852800,   2995809750614016,  1456296406548480,   475525357240320,  100306130042880,   12383472844800,    681091006464;
  9: 1521681143169024, 16738492574859264, 62769347155722240, 125538694311444480, 156923367889305600, 131815629027016704, 76892450265759744, 31384673577861120, 8826939443773440, 1634618415513600, 179808025706496, 8916100448256;

The row polynomials satisfy (2+(n+2)*x)*P(n,x) = (48+(24*n+96)*x+(12*n+36)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 1152*x + 432*x^2.

P[0, x_] := 576 + 1152*x + 432*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((48 + (24*n + 96)*x + (12*n + 36)*x^2)*P[n - 1, x])/(2 + (n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 576 + 1152*x + 432*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((48 + (24*n + 96)*x + (12*n + 36)*x^2)*v[i - 1])/(2 + (n + 2)*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)*12^(i-j)*12^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^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)*12^(i-j)*12^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A113214, shifted → x ↦ x+1 → θ+1
• A038207 → θ+1
• A178820 → basis A007318 → θ+1
• A038330 → basis A013609 → θ+1
• A120910 → θ+1
• A178822 → x ↦ x+1 → θ+1
• A196347 → x ↦ x+1 → θ+1
• A356546 → x ↦ x+1 → θ+1
• image under θ+1 → θ+1 → D+1 → basis A013609
• A038221 → x ↦ x+1 → θ+1
• A085880 → x ↦ x+1 → θ+1
• A131431 → D+1 → basis A013609
• A085881 → basis A007318 → θ+1
• A132440 → D+1 → basis A013609
• A120910 → D → θ+1
• A003506 → x ↦ x+1 → θ+1
• A027466 → basis A013611 → θ+1
• A038208 → basis A007318 → θ+1
• A178821 → basis A007318 → θ+1
• A028326 → basis A007318 → θ+1
• A094305 → x ↦ x+1 → θ+1
• A156992 → basis A007318 → θ+1
• A135065 → x ↦ x+1 → θ+1
• A038325 → basis A007318 → θ+1

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.