A038328 → basis A013609 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:            324,             144,              16;
  1:           5832,            3888,             864,              64;
  2:         104976,           93312,           31104,            4608,            256;
  3:        1889568,         2099520,          933120,          207360,          23040,           1024;
  4:       34012224,        45349632,        25194240,         7464960,        1244160,         110592,          4096;
  5:      612220032,       952342272,       634894848,       235146240,       52254720,        6967296,        516096,        16384;
  6:    11019960576,     19591041024,     15237476352,      6772211712,     1881169920,      334430208,      37158912,      2359296,       65536;
  7:   198359290368,    396718580736,    352638738432,    182849716224,    60949905408,    13544423424,    2006581248,    191102976,    10616832,     262144;
  8:  3570467226624,   7934371614720,   7934371614720,   4701849845760,  1828497162240,   487599243264,   90296156160,  11466178560,   955514880,   47185920,   1048576;
  9: 64268410079232, 157100557971456, 174556175523840, 116370783682560, 51720348303360, 16090775027712, 3575727783936, 567575838720, 63063982080, 4671406080, 207618048, 4194304;

The row polynomials satisfy P(n,x) = (18+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 144*x + 16*x^2.

P[0, x_] := 324 + 144*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(18 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 324 + 144*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 4*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 (−∞, -1]
  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), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013609 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 basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A038279 → basis A013609 → D
• A038279 → basis A013609
• A027467 → basis A013609 → basis A013609
• A038279 → basis A013609 → reverse

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.