A038336 → (θ+1)²

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Submission text ?

Triangle begins:
  0:          144,            960,             900;
  1:         1728,          17280,           32400,            16000;
  2:        20736,         276480,          777600,           768000,           250000;
  3:       248832,        4147200,        15552000,         23040000,         15000000,          3600000;
  4:      2985984,       59719680,       279936000,        552960000,        540000000,        259200000,         49000000;
  5:     35831808,      836075520,      4702924800,      11612160000,      15120000000,      10886400000,       4116000000,        640000000;
  6:    429981696,    11466178560,     75246796800,     222953472000,     362880000000,     348364800000,     197568000000,      61440000000,       8100000000;
  7:   5159780352,   154793410560,   1160950579200,    4013162496000,    7838208000000,    9405849600000,    7112448000000,    3317760000000,     874800000000,    100000000000;
  8:  61917364224,  2063912140800,  17414258688000,   68797071360000,  156764160000000,  225740390400000,  213373440000000,  132710400000000,   52488000000000,  12000000000000,   1210000000000;
  9: 743008370688, 27243640258560, 255409127424000, 1135151677440000, 2956124160000000, 4966288588800000, 5633058816000000, 4379443200000000, 2309472000000000, 792000000000000, 159720000000000, 14400000000000;

The row polynomials satisfy (36+(90*n+150)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (432+(1080*n+3240)*x+(300*n^2+2700*n+5100)*x^2+(250*n^2+1500*n+2250)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 960*x + 900*x^2.

P[0, x_] := 144 + 960*x + 900*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((432 + (1080*n + 3240)*x + (300*n^2 + 2700*n + 5100)*x^2 + (250*n^2 + 1500*n + 2250)*x^3)*P[n - 1, x])/(36 + (90*n + 150)*x + (25*n^2 + 100*n + 100)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 960*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((432 + (1080*n + 3240)*x + (300*n^2 + 2700*n + 5100)*x^2 + (250*n^2 + 1500*n + 2250)*x^3)*v[i - 1])/(36 + (90*n + 150)*x + (25*n^2 + 100*n + 100)*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)*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 [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 (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Parent-row provers of polytool (above)
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A013612 → x ↦ x+1 → (θ+1)²
• A038271 → basis A013609 → (θ+1)²
• A038259 → (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.