A038328 → reverse

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Submission text ?

Triangle begins:
  0:    4,     48,     144;
  1:    8,    144,     864,     1728;
  2:   16,    384,    3456,    13824,     20736;
  3:   32,    960,   11520,    69120,    207360,     248832;
  4:   64,   2304,   34560,   276480,   1244160,    2985984,     2985984;
  5:  128,   5376,   96768,   967680,   5806080,   20901888,    41803776,     35831808;
  6:  256,  12288,  258048,  3096576,  23224320,  111476736,   334430208,    573308928,    429981696;
  7:  512,  27648,  663552,  9289728,  83607552,  501645312,  2006581248,   5159780352,   7739670528,    5159780352;
  8: 1024,  61440, 1658880, 26542080, 278691840, 2006581248, 10032906240,  34398535680,  77396705280,  103195607040,   61917364224;
  9: 2048, 135168, 4055040, 72990720, 875888640, 7357464576, 44144787456, 189191946240, 567575838720, 1135151677440, 1362182012928, 743008370688;

The row polynomials satisfy P(n,x) = (2+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 48*x + 144*x^2.

P[0, x_] := 4 + 48*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 4 + 48*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 12*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)*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 ?

Same fingerprint ?

• A013613: Triangle of coefficients in expansion of (1+6x)^n.
• A013613 → D
• A038255 → D
• A038255: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).
• A147716 → basis A013610 → x ↦ x+1
• A038328 → D
• A038291 → basis A013609 → basis A007318
• A038329 → x ↦ x+1 → x ↦ x+1
• A013612 → x ↦ x/(x+1)
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.

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.