A013612 → basis A013609 → x ↦ x/(x+1)

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Submission text ?

Triangle begins:
  0:        36,         192,          256;
  1:       216,        1728,         4608,          4096;
  2:      1296,       13824,        55296,         98304,         65536;
  3:      7776,      103680,       552960,       1474560,       1966080,        1048576;
  4:     46656,      746496,      4976640,      17694720,      35389440,       37748736,       16777216;
  5:    279936,     5225472,     41803776,     185794560,     495452160,      792723456,      704643072,       268435456;
  6:   1679616,    35831808,    334430208,    1783627776,    5945425920,    12683575296,    16911433728,     12884901888,      4294967296;
  7:  10077696,   241864704,   2579890176,   16052649984,   64210599936,   171228266496,   304405807104,    347892350976,    231928233984,     68719476736;
  8:  60466176,  1612431360,  19349176320,  137594142720,  642105999360,  2054739197952,  4566087106560,   6957847019520,   6957847019520,   4123168604160,  1099511627776;
  9: 362797056, 10642046976, 141893959680, 1135151677440, 6054142279680, 22602131177472, 60272349806592, 114804475822080, 153072634429440, 136064563937280, 72567767433216, 17592186044416;

The row polynomials satisfy (1+x)*P(n,x) = (6+22*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 192*x + 256*x^2.

P[0, x_] := 36 + 192*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6 + 22*x + 16*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 36 + 192*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6 + 22*x + 16*x^2)*v[i - 1])/(1 + x)); 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, 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_A013609 of Triangle of coefficients in expansion of (1+5x)^n.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (1+5x)^n.?
        • 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 ?

• A027466 → basis A013610
• Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
• A013609 → basis A013610 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013611
• A028326 → basis A013611 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013611
• A178821 → basis A013611 → basis A013611
• A038281 → D
• A013622 → x ↦ x/(x+1) → D
• A120909 → basis A013610 → basis A013611
• A013609 → D → basis A013611
• A038327 → basis A013610 → basis A013609
• A027466 → basis A013610 → D
• A013622 → x ↦ x/(x+1)
• A136572 → basis A013609 → basis A013611
• A130123 → basis A013609 → basis A013611
• A120909 → basis A013611
• basis-map coefficients of A112227 → basis A013609 → basis A013611
• A038328 → x ↦ x+1 → basis A013610
• A038208 → basis A013611 → basis A013611
• A178820 → basis A013611 → basis A013611
• basis-map coefficients of A026729 → basis A013609 → basis A013611
• A038271 → basis A013611 → x ↦ x/(x+1)
• A038226 → D
• A038329 → basis A013610 → basis A007318
• A038281: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^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.