A038331 → basis A013610 → reverse

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Submission text ?

Triangle begins:
  0:           225,             510,             289;
  1:          3375,           11475,           13005,             4913;
  2:         50625,          229500,          390150,           294780,            83521;
  3:        759375,         4303125,         9753750,         11054250,          6264075,          1419857;
  4:      11390625,        77456250,       219459375,        331627500,        281883375,        127787130,         24137569;
  5:     170859375,      1355484375,      4608646875,       8705221875,       9865918125,       6708824325,       2534444745,        410338673;
  6:    2562890625,     23236875000,     92172937500,     208925325000,     295977543750,     268352973000,     152066684700,      49240640760,       6975757441;
  7:   38443359375,    392122265625,   1777620937500,    4700819812500,    7991393681250,    9056912838750,    6843000811500,    3323743251300,     941727254535,     118587876497;
  8:  576650390625,   6535371093750,  33330392578125,  100731853125000,  199784842031250,  271707385162500,  256612530431250,  166187162565000,   70629544090125,   17788181474550,   2015993900449;
  9: 8649755859375, 107833623046875, 611057197265625, 2077594470703125, 4709214133593750, 7471953091968750, 8468213504231250, 6855220455806250, 3884624924956875, 1467524971650375, 332638993574085, 34271896307633;

The row polynomials satisfy P(n,x) = (15+17*x)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 510*x + 289*x^2.

P[0, x_] := 225 + 510*x + 289*x^2;
P[n_, x_] := P[n, x] = Expand[(15 + 17*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 225 + 510*x + 289*x^2; for(n = 1, N, my(i = n + 1); v[i] = (15 + 17*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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • 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
        • 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

Related ?

Same fingerprint ?

• A038331 → basis A013610
• A038271 → basis A007318 → basis A013610
• A038331 → basis A013610 → D
• A147716 → basis A013609 → 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.