A139548 → x ↦ x+1 → reverse

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Submission text ?

Triangle begins:
  0:   1;
  1:   2,     8,      8;
  2:   4,    32,     96,     128,       64;
  3:   8,    96,    480,    1280,     1920,      1536,       512;
  4:  16,   256,   1792,    7168,    17920,     28672,     28672,      16384,       4096;
  5:  32,   640,   5760,   30720,   107520,    258048,    430080,     491520,     368640,      163840,       32768;
  6:  64,  1536,  16896,  112640,   506880,   1622016,   3784704,    6488064,    8110080,     7208960,     4325376,     1572864,      262144;
  7: 128,  3584,  46592,  372736,  2050048,   8200192,  24600576,   56229888,   98402304,   131203072,   131203072,    95420416,    47710208,    14680064,     2097152;
  8: 256,  8192, 122880, 1146880,  7454720,  35782656, 131203072,  374865920,  843448320,  1499463680,  2099249152,  2290089984,  1908408320,  1174405120,   503316480,   134217728,   16777216;
  9: 512, 18432, 313344, 3342336, 25067520, 140378112, 608305152, 2085617664, 5735448576, 12745441280, 22941794304, 33369882624, 38931529728, 35936796672, 25669140480, 13690208256, 5133828096, 1207959552, 134217728;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + 8*x + 8*x^2)*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_A007318 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
            • 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
            • 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 (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Candidate equalities ?

• A139548 → x ↦ x/(x+1) 15 rows; no all-index equality certificate was found

Same fingerprint ?

• A034870 → basis A013611
• A034870 → basis A013611 → basis A013610
• A034870 → x ↦ x/(x+1)
• A139548 → x ↦ x+1
• basis-map coefficients of A297191, shifted → θ+1 → basis A013609
• basis-map coefficients of A297191, shifted → basis A013609
• basis-map coefficients of A297191, shifted → (θ+1)² → basis A013609
• A139548 → basis A013611
• A034870 → x ↦ x+1
• basis-map coefficients of A297191, shifted → basis A013609 → basis A013610
• A139548 → basis A013609 → basis A007318
• A034870 → basis A013609 → x ↦ x+1
• A034870 → basis A013609 → basis A013611

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.