A038328 → x ↦ x+1 → D

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Submission text ?

Triangle begins:
  0:            56,             8;
  1:          1176,           336,            24;
  2:         21952,          9408,          1344,            64;
  3:        384160,        219520,         47040,          4480,          160;
  4:       6453888,       4609920,       1317120,        188160,        13440,         384;
  5:     105413504,      90354432,      32269440,       6146560,       658560,       37632,         896;
  6:    1686616064,    1686616064,     722835456,     172103680,     24586240,     2107392,      100352,      2048;
  7:   26564203008,   30359089152,   15179544576,    4337012736,    774466560,    88510464,     6322176,    258048,     4608;
  8:  413220935680,  531284060160,  303590891520,  101196963840,  21685063680,  3097866240,   295034880,  18063360,   645120,   10240;
  9: 6363602409472, 9090860584960, 5844124661760, 2226333204480, 556583301120, 95414280192, 11358842880, 927252480, 49674240, 1576960, 22528;

The row polynomials satisfy (n+1)*P(n,x) = (14*n+28+(2*n+4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 56 + 8*x.

P[0, x_] := 56 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((14*n + 28 + (2*n + 4)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 56 + 8*x; for(n = 1, N, my(i = n + 1); v[i] = ((14*n + 28 + (2*n + 4)*x)*v[i - 1])/(n + 1)); 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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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)*2^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A027466: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).
• A038328 → x ↦ x+1
• A038327 → x ↦ x+1 → basis A013609
• A027466 → reverse
• A013613 → x ↦ x/(x+1)
• A038328 → x ↦ x+1 → reverse
• A027466 → D

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.