A038330 → basis A013610 → θ+1

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Submission text ?

Triangle begins:
  0:            256,             768,              432;
  1:           4096,           18432,            20736,             6912;
  2:          65536,          393216,           663552,           442368,           103680;
  3:        1048576,         7864320,         17694720,         17694720,          8294400,           1492992;
  4:       16777216,       150994944,        424673280,        566231040,        398131200,         143327232,          20901888;
  5:      268435456,      2818572288,       9512681472,      15854469120,      14863564800,        8026324992,        2341011456,        286654464;
  6:     4294967296,     51539607552,     202937204736,     405874409472,     475634073600,      342456532992,      149824733184,      36691771392,       3869835264;
  7:    68719476736,    927712935936,    4174708211712,    9740985827328,   13698261319680,    12328435187712,     7191587192832,    2641807540224,     557256278016,     51597803520;
  8:  1099511627776,  16492674416640,   83494164234240,  222651104624640,  365286968524800,   394509926006784,   287663487713280,  140896402145280,   44580502241280,   8255648563200,    681091006464;
  9: 17592186044416, 290271069732864, 1632774767247360, 4898324301742080, 9184358065766400, 11572291162865664, 10125754767507456, 6199441694392320, 2615389464821760, 726497073561600, 119872017137664, 8916100448256;

The row polynomials satisfy (4+(3*n+6)*x)*P(n,x) = (64+(48*n+192)*x+(36*n+108)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 768*x + 432*x^2.

P[0, x_] := 256 + 768*x + 432*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((64 + (48*n + 192)*x + (36*n + 108)*x^2)*P[n - 1, x])/(4 + (3*n + 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 256 + 768*x + 432*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((64 + (48*n + 192)*x + (36*n + 108)*x^2)*v[i - 1])/(4 + (3*n + 6)*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 (−∞, 0]
  by Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
        • 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 ?

• A038284 → θ+1
• A013610 → basis A007318 → θ+1
• A038335 → θ+1

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.