A038333 → basis A013609 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:              1089,                924,                196;
  1:             35937,              45738,              19404,               2744;
  2:           1185921,            2012472,            1280664,             362208,              38416;
  3:          39135393,           83014470,           70436520,           29882160,            6338640,             537824;
  4:        1291467969,         3287373012,         3486607740,         1972222560,          627525360,          106489152,            7529536;
  5:       42618442977,       126563860962,       161081277588,       113895852840,        48319452720,        12299497056,         1739322816,         105413504;
  6:     1406408618241,      4773265613424,      7087576213872,      6013701029952,      3189083879520,      1082355740928,       229590611712,       27829165056,       1475789056;
  7:    46411484401953,    177207485898366,    300715733645712,    297678200982624,    189431582443488,     80364913763904,     22729470559488,     4132631010816,     438309349632,      20661046784;
  8:  1531578985264449,   6497607816273420,  12404524012885620,  14033400903466560,  10418737034391840,   5304084308417664,   1875181321157760,   454589411189760,   72321042689280,    6818145438720,    289254654976;
  9: 50542106513726817, 235863163730725146, 500315801853053340, 636765565994795160, 540285934783462560, 320897100659268672, 136138163916053376, 41253989065470720, 8750846165402880, 1237493397127680, 104999439756288, 4049565169664;

The row polynomials satisfy P(n,x) = (33+14*x)*P(n-1,x) for n >= 1, with P(0,x) = 1089 + 924*x + 196*x^2.

P[0, x_] := 1089 + 924*x + 196*x^2;
P[n_, x_] := P[n, x] = Expand[(33 + 14*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1089 + 924*x + 196*x^2; for(n = 1, N, my(i = n + 1); v[i] = (33 + 14*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 (−∞, -1]
  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), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013609 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 basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.?
    • ↑ Geometric basis map (elementary) (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.