A038328 → basis A013611 → reverse

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Submission text ?

Triangle begins:
  0:         64,          224,           196;
  1:        512,         2688,          4704,          2744;
  2:       4096,        28672,         75264,         87808,          38416;
  3:      32768,       286720,       1003520,       1756160,        1536640,         537824;
  4:     262144,      2752512,      12042240,      28098560,       36879360,       25815552,         7529536;
  5:    2097152,     25690112,     134873088,     393379840,      688414720,      722835456,       421654016,       105413504;
  6:   16777216,    234881024,    1438646272,    5035261952,    11014635520,    15420489728,     13492928512,      6746464256,      1475789056;
  7:  134217728,   2113929216,   14797504512,   60423143424,   158610751488,   277568815104,    323830284288,    242872713216,    106256812032,    20661046784;
  8: 1073741824,  18790481920,  147975045120,  690550210560,  2114810019840,  4441101041664,   6476605685760,   6476605685760,   4250272481280,  1652883742720,   289254654976;
  9: 8589934592, 165356240896, 1446867107840, 7596052316160, 26586183106560, 65136148611072, 113988260069376, 142485325086720, 124674659450880, 72726884679680, 25454409637888, 4049565169664;

The row polynomials satisfy P(n,x) = (8+14*x)*P(n-1,x) for n >= 1, with P(0,x) = 64 + 224*x + 196*x^2.

P[0, x_] := 64 + 224*x + 196*x^2;
P[n_, x_] := P[n, x] = Expand[(8 + 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] = 64 + 224*x + 196*x^2; for(n = 1, N, my(i = n + 1); v[i] = (8 + 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 (−∞, 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_A013611 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

Related ?

Candidate equalities ?

• A038284 → x ↦ x/(x+1) 15 rows; no all-index equality certificate was found

Same fingerprint ?

• A038291 → basis A013611 → basis A013609
• A038255 → basis A013611
• A038329 → basis A013609 → basis A013609
• A120910 → basis A013611 → x ↦ x+1
• A038255 → basis A013611 → reverse
• A038255 → basis A013611 → D
• A013610 → basis A007318 → x ↦ x/(x+1)
• A038335 → x ↦ x/(x+1)
• A038334 → x ↦ x+1 → basis A013609
• A038328 → basis A013611
• A038328 → basis A013611 → D
• A038330 → basis A013610 → 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.