A038338 → basis A013610 → basis A013610

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Submission text ?

Triangle begins:
  0:                 3600,                 12960,                  11664;
  1:               216000,               1166400,                2099520,                 1259712;
  2:             12960000,              93312000,              251942400,               302330880,                136048896;
  3:            777600000,            6998400000,            25194240000,             45349632000,              40814668800,              14693280768;
  4:          46656000000,          503884800000,          2267481600000,           5441955840000,            7346640384000,            5289581076480,            1586874322944;
  5:        2799360000000,        35271936000000,        190468454400000,         571405363200000,         1028529653760000,         1110812026060800,          666487215636480,          171382426877952;
  6:      167961600000000,      2418647040000000,      15237476352000000,       54854914867200000,       123423558451200000,       177729924169728000,       159956931752755200,        82263564901416960,        18509302102818816;
  7:    10077696000000000,    163258675200000000,    1175462461440000000,     4936942338048000000,     13329744312729600000,     23993539762913280000,     28792247715495936000,     22211162523382579200,      9995023135522160640,      1999004627104432128;
  8:   604661760000000000,  10883911680000000000,   88159684608000000000,   423166486118400000000,   1332974431272960000000,   2879224771549593600000,   4318837157324390400000,   4442232504676515840000,   2998506940656648192000,   1199402776262659276800,    215892499727278669824;
  9: 36279705600000000000, 718338170880000000000, 6465043537920000000000, 34911235104768000000000, 125680446377164800000000, 316714724870455296000000, 570086504766819532800000, 732968363271625113600000, 659671526944462602240000, 395802916166677561344000, 142489049820003922083840, 23316389970546096340992;

The row polynomials satisfy P(n,x) = (60+108*x)*P(n-1,x) for n >= 1, with P(0,x) = 3600 + 12960*x + 11664*x^2.

P[0, x_] := 3600 + 12960*x + 11664*x^2;
P[n_, x_] := P[n, x] = Expand[(60 + 108*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 3600 + 12960*x + 11664*x^2; for(n = 1, N, my(i = n + 1); v[i] = (60 + 108*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)
• 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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^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)*12^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A028326 → basis A013610 → basis A013610
• A156992 → basis A013610 → basis A013610
• A178821 → basis A013610 → basis A013610
• A038271 → basis A013610 → basis A007318
• A013622 → basis A013609 → basis A007318
• A038319 → basis A013611 → x ↦ x+1
• A013620 → basis A013610 → reverse
• A038208 → basis A013610 → basis A013610
• A038323 → basis A013611
• A038334 → basis A013609 → x ↦ x/(x+1)
• A038329 → basis A013611 → x ↦ x/(x+1)
• A196347 → basis A013610 → basis A013610
• A085880 → basis A013610 → basis A013610
• A178822 → basis A013610 → basis A013610
• A038221 → basis A013610 → basis A013610
• A013620 → basis A013610
• A178820 → basis A013610 → basis A013610
• A013620 → basis A013610 → D

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.