A038333 → x ↦ x+1 → reverse

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Submission text ?

Triangle begins:
  0:         49,         266,          361;
  1:        343,        2793,         7581,          6859;
  2:       2401,       26068,       106134,        192052,         130321;
  3:      16807,      228095,      1238230,       3360910,        4561235,         2476099;
  4:     117649,     1915998,     13001415,      47052740,       95785935,       103996158,        47045881;
  5:     823543,    15647317,    127413867,     576396065,     1564503605,      2547905871,      2305248169,       893871739;
  6:    5764801,   125178536,   1189196092,    6455635928,    21903050470,     47560909592,     64546948732,     50056817384,     16983563041;
  7:   40353607,   985780971,  10702764828,   67784177244,   275978435922,    749084326074,   1355485923372,   1576789747596,   1069964471583,    322687697779;
  8:  282475249,  7667185330,  93649192245,  677841772440,  3219748419090,  10487180565036,  23721003659010,  36791760777240,  37448756505405,  22588138844530,   6131066257801;
  9: 1977326743, 59037327041, 801220866985, 6524227059735, 35417232609990, 134585483917962, 365303456348754, 708241394961870, 961184750305395, 869643345514405, 472092101850677, 116490258898219;

The row polynomials satisfy P(n,x) = (7+19*x)*P(n-1,x) for n >= 1, with P(0,x) = 49 + 266*x + 361*x^2.

P[0, x_] := 49 + 266*x + 361*x^2;
P[n_, x_] := P[n, x] = Expand[(7 + 19*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 49 + 266*x + 361*x^2; for(n = 1, N, my(i = n + 1); v[i] = (7 + 19*x)*v[i - 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 Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(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)*12^(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)

Related ?

Same fingerprint ?

• A038333 → x ↦ x+1
• A038271 → x ↦ x/(x+1) → x ↦ x/(x+1)
• A038333 → x ↦ x+1 → D
• A038284 → basis A013611 → x ↦ x/(x+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.