A038328 → x ↦ x+1 → basis A013610

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Submission text ?

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

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 256 + 192*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 6*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 Geometric basis map (elementary), Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A007318 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
    • 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) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A112227 → basis A013609 → basis A013611
• A038281: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
• A013609 → D → basis A013611
• A038226 → D
• A038208 → basis A013611 → basis A013611
• Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
• A027466 → basis A013610 → D
• A120909 → basis A013611
• A038281 → D
• A027466 → basis A013610
• A178821 → basis A013611 → basis A013611
• A038271 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013611
• A013612 → basis A013609 → x ↦ x/(x+1)
• A178820 → basis A013611 → basis A013611
• A013622 → x ↦ x/(x+1) → D
• A130123 → basis A013609 → basis A013611
• A038329 → basis A013610 → basis A007318
• A120909 → basis A013610 → basis A013611
• A013622 → x ↦ x/(x+1)
• A013609 → basis A013610 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013611
• A136572 → basis A013609 → basis A013611
• basis-map coefficients of A026729 → basis A013609 → basis A013611
• A028326 → basis A013611 → basis A013611
• A038327 → basis A013610 → basis A013609

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.