A038314 → (θ+1)²

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Submission text ?

Triangle begins:
  0:          100,           960,           1296;
  1:         1000,         14400,          38880,           27648;
  2:        10000,        192000,         777600,         1105920,           518400;
  3:       100000,       2400000,       12960000,        27648000,         25920000,          8957952;
  4:      1000000,      28800000,      194400000,       552960000,        777600000,        537477120,        146313216;
  5:     10000000,     336000000,     2721600000,      9676800000,      18144000000,      18811699200,      10241925120,       2293235712;
  6:    100000000,    3840000000,    36288000000,    154828800000,     362880000000,     501645312000,     409677004800,     183458856960,      34828517376;
  7:   1000000000,   43200000000,   466560000000,   2322432000000,    6531840000000,   11287019520000,   12290310144000,    8255648563200,    3134566563840,     515978035200;
  8:  10000000000,  480000000000,  5832000000000,  33177600000000,  108864000000000,  225740390400000,  307257753600000,  275188285440000,  156728328192000,   51597803520000,   7492001071104;
  9: 100000000000, 5280000000000, 71280000000000, 456192000000000, 1710720000000000, 4138573824000000, 6759670579200000, 7567677849600000, 5746705367040000, 2837879193600000, 824120117821440, 106993205379072;

The row polynomials satisfy (25+(90*n+150)*x+(36*n^2+144*n+144)*x^2)*P(n,x) = (250+(900*n+2700)*x+(360*n^2+3240*n+6120)*x^2+(432*n^2+2592*n+3888)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 100 + 960*x + 1296*x^2.

P[0, x_] := 100 + 960*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((250 + (900*n + 2700)*x + (360*n^2 + 3240*n + 6120)*x^2 + (432*n^2 + 2592*n + 3888)*x^3)*P[n - 1, x])/(25 + (90*n + 150)*x + (36*n^2 + 144*n + 144)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 100 + 960*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((250 + (900*n + 2700)*x + (360*n^2 + 3240*n + 6120)*x^2 + (432*n^2 + 2592*n + 3888)*x^3)*v[i - 1])/(25 + (90*n + 150)*x + (36*n^2 + 144*n + 144)*x^2)); 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 Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^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 Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

• A038220 → basis A013610 → (θ+1)²
• A038334 → basis A013610 → (θ+1)²

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.