A038331 → basis A013611 → basis A007318

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Submission text ?

Triangle begins:
  0:               1369,                1480,                 400;
  1:              50653,               82140,               44400,                8000;
  2:            1874161,             4052240,             3285600,             1184000,              160000;
  3:           69343957,           187416100,           202612000,           109520000,            29600000,             3200000;
  4:         2565726409,          8321274840,         11244966000,          8104480000,          3285600000,           710400000,            64000000;
  5:        94931877133,        359201697260,        582489238800,        524765080000,        283656800000,         91996800000,         16576000000,         1280000000;
  6:      3512479453921,      15189100341280,      28736135780800,      31066092736000,      20990603200000,       9077017600000,       2453248000000,       378880000000,        25600000000;
  7:    129961739795077,     632246301705780,    1367019030715200,    1724168146848000,    1397974173120000,     755661715200000,     272310528000000,     63083520000000,      8524800000000,      512000000000;
  8:   4808584372417849,   25992347959015400,   63224630170578000,   91134602047680000,   86208407342400000,   55918966924800000,   25188723840000000,   7780300800000000,   1577088000000000,   189440000000000,   10240000000000;
  9: 177917621779460413, 1057888561931926780, 2859158275491694000, 4636472879175720000, 5012403112622400000, 3793169923065600000, 2050362120576000000, 791645606400000000, 213958272000000000, 38551040000000000, 4167680000000000, 204800000000000;

The row polynomials satisfy P(n,x) = (37+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 1369 + 1480*x + 400*x^2.

P[0, x_] := 1369 + 1480*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[(37 + 20*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1369 + 1480*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (37 + 20*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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (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_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^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
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    • ↑ Geometric basis map (elementary) (above)

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.