A034870 → basis A013609 → basis A013609

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Submission text ?

Triangle begins:
  0:           1;
  1:          16,            32,             16;
  2:         256,          1024,           1536,           1024,             256;
  3:        4096,         24576,          61440,          81920,           61440,           24576,             4096;
  4:       65536,        524288,        1835008,        3670016,         4587520,         3670016,          1835008,           524288,            65536;
  5:     1048576,      10485760,       47185920,      125829120,       220200960,       264241152,        220200960,        125829120,         47185920,         10485760,          1048576;
  6:    16777216,     201326592,     1107296256,     3690987520,      8304721920,     13287555072,      15502147584,      13287555072,       8304721920,       3690987520,       1107296256,        201326592,         16777216;
  7:   268435456,    3758096384,    24427626496,    97710505984,    268703891456,    537407782912,     806111674368,     921270484992,     806111674368,     537407782912,     268703891456,      97710505984,      24427626496,      3758096384,       268435456;
  8:  4294967296,   68719476736,   515396075520,  2405181685760,   7816840478720,  18760417148928,   34394098106368,   49134425866240,   55276229099520,   49134425866240,   34394098106368,   18760417148928,    7816840478720,   2405181685760,    515396075520,    68719476736,     4294967296;
  9: 68719476736, 1236950581248, 10514079940608, 56075093016576, 210281598812160, 588788476674048, 1275708366127104, 2186928627646464, 3007026863013888, 3341140958904320, 3007026863013888, 2186928627646464, 1275708366127104, 588788476674048, 210281598812160, 56075093016576, 10514079940608, 1236950581248, 68719476736;

The row polynomials satisfy P(n,x) = (16+32*x+16*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[(16 + 32*x + 16*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] = (16 + 32*x + 16*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 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)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013609 of Even-numbered rows of Pascal's triangle.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Even-numbered rows of Pascal's triangle.?
        • 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 Even-numbered rows of Pascal's triangle.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • Geometric basis map (elementary) polytool ? step ›

              • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                • 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 basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Iterated real-rootedness preservers (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • 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
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of Even-numbered rows of Pascal's triangle.?
        • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A139548 → x ↦ x+1 → basis A013610
• A139548 → basis A013609 → basis A013609
• A139548: 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.
• basis-map coefficients of A297191, shifted → (θ+1)² → basis A007318
• A034870 → D → D
• A034870 → x ↦ x+1 → basis A013610
• A139548 → basis A013609
• A034870 → basis A013609
• basis-map coefficients of A297191, shifted → θ+1 → x ↦ x+1
• A034870: Even-numbered rows of Pascal's triangle.

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.