A038284 → reverse

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Submission text ?

Triangle begins:
  0:        36,         96,          64;
  1:       216,        864,        1152,          512;
  2:      1296,       6912,       13824,        12288,         4096;
  3:      7776,      51840,      138240,       184320,       122880,        32768;
  4:     46656,     373248,     1244160,      2211840,      2211840,      1179648,       262144;
  5:    279936,    2612736,    10450944,     23224320,     30965760,     24772608,     11010048,      2097152;
  6:   1679616,   17915904,    83607552,    222953472,    371589120,    396361728,    264241152,    100663296,     16777216;
  7:  10077696,  120932352,   644972544,   2006581248,   4013162496,   5350883328,   4756340736,   2717908992,    905969664,    134217728;
  8:  60466176,  806215680,  4837294080,  17199267840,  40131624960,  64210599936,  71345111040,  54358179840,  27179089920,   8053063680,  1073741824;
  9: 362797056, 5321023488, 35473489920, 141893959680, 378383892480, 706316599296, 941755465728, 896909967360, 597939978240, 265751101440, 70866960384, 8589934592;

The row polynomials satisfy P(n,x) = (6+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 96*x + 64*x^2.

P[0, x_] := 36 + 96*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(6 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 36 + 96*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 8*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 Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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 ?

Same fingerprint ?

• basis-map coefficients of A026729, shifted → basis A013609 → basis A013609
• A038330 → basis A013610 → D
• A178821 → basis A013611 → basis A013609
• basis-map coefficients of A026729 → basis A013610 → basis A007318
• A027465 → x ↦ x/(x+1)
• A178822 → basis A013611 → basis A013609
• A196347 → x ↦ x+1 → basis A013611
• A085880 → x ↦ x+1 → basis A013611
• A013609 → basis A013609
• A003506 → x ↦ x+1 → basis A013611
• A013610 → basis A013611 → x ↦ x+1
• A130123 → basis A013610 → x ↦ x+1
• A038279 → basis A013610 → x ↦ x/(x+1)
• A038243 → basis A013609 → x ↦ x/(x+1)
• basis-map coefficients of A030528, shifted → basis A013610 → x ↦ x+1
• A120910 → D → basis A013611
• Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013609
• A130123 → basis A013609 → basis A013609
• A038284 → D
• A038335 → reverse
• A178820 → basis A007318 → basis A013611
• A038330 → basis A013610
• A120910 → basis A013611 → reverse
• A356546 → x ↦ x+1 → basis A013611
• A178821 → basis A007318 → basis A013611
• A085880 → basis A013611 → basis A013609
• A094305 → x ↦ x+1 → basis A013611
• A038334 → basis A013609 → basis A013610
• A120910 → basis A013611
• A356546 → basis A013611 → basis A013609
• A178820 → basis A013611 → basis A013609
• A013609 → basis A013609 → D
• A038335 → D
• A120910 → basis A013611 → D
• basis-map coefficients of A112227 → basis A013609 → basis A013609
• A013609 → D → basis A013609
• A156992 → basis A013611 → basis A013609
• A038330 → basis A013610 → reverse
• A038330 → x ↦ x/(x+1)
• basis-map coefficients of A026729 → basis A013609 → basis A013609
• A028326 → basis A007318 → basis A013611
• A038208 → basis A013611 → basis A013609
• A038221 → basis A013611 → basis A013609
• A196347 → basis A013611 → basis A013609
• basis-map coefficients of A026729, shifted → basis A013610 → x ↦ x+1
• A028326 → basis A013611 → basis A013609
• basis-map coefficients of A112227 → basis A013610 → basis A007318
• A136572 → basis A013610 → basis A007318
• A085881 → basis A013611 → basis A013609
• A013610 → basis A007318 → D
• A178822 → x ↦ x+1 → basis A013611
• A027465 → D → basis A013610
• A038335: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
• A038329 → basis A013611 → basis A013610
• A135065 → x ↦ x+1 → basis A013611
• A120909 → basis A013610 → basis A013609
• A038329 → basis A013610 → basis A013609
• A013610 → basis A007318

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.