A013612 → basis A013611 → D

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Submission text ?

Triangle begins:
  0:         240,          800;
  1:        2160,        14400,         24000;
  2:       17280,       172800,        576000,         640000;
  3:      129600,      1728000,       8640000,       19200000,        16000000;
  4:      933120,     15552000,     103680000,      345600000,       576000000,        384000000;
  5:     6531840,    130636800,    1088640000,     4838400000,     12096000000,      16128000000,       8960000000;
  6:    44789760,   1045094400,   10450944000,    58060800000,    193536000000,     387072000000,     430080000000,     204800000000;
  7:   302330880,   8062156800,   94058496000,   627056640000,   2612736000000,    6967296000000,   11612160000000,   11059200000000,    4608000000000;
  8:  2015539200,  60466176000,  806215680000,  6270566400000,  31352832000000,  104509440000000,  232243200000000,  331776000000000,  276480000000000,  102400000000000;
  9: 13302558720, 443418624000, 6651279360000, 59122483200000, 344881152000000, 1379524608000000, 3832012800000000, 7299072000000000, 9123840000000000, 6758400000000000, 2252800000000000;

The row polynomials satisfy (n+1)*P(n,x) = (6*n+12+(20*n+40)*x)*P(n-1,x) for n >= 1, with P(0,x) = 240 + 800*x.

P[0, x_] := 240 + 800*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 12 + (20*n + 40)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 240 + 800*x; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 12 + (20*n + 40)*x)*v[i - 1])/(n + 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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013611 of Triangle of coefficients in expansion of (1+5x)^n.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (1+5x)^n.?
        • 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 ?

• A038291 → basis A013610
• A038284 → basis A007318 → x ↦ x+1
• A038305 → reverse
• A038335 → x ↦ x+1 → basis A007318
• A038305 → D
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.
• A038291 → basis A013610 → D
• A013612 → basis A013611
• A038291 → basis A013610 → 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.