A038325 → basis A013609 → (θ+1)²

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Triangle begins:
  0:             1;
  1:            22,             88;
  2:           484,           3872,            4356;
  3:         10648,         127776,          287496,           170368;
  4:        234256,        3748096,        12649824,         14992384,          5856400;
  5:       5153632,      103072640,       463826880,        824581120,        644204000,        185530752;
  6:     113379904,     2721117696,     15306287040,      36281569280,      42517464000,      24490059264,       5555615296;
  7:    2494357888,    69842020864,    471433640832,    1396840417280,    2182563152000,    1885734563328,     855564755584,     159638904832;
  8:   54875873536,  1756027953152,  13828720131072,   49168782688256,   96032778688000,  110629761048576,   75289698491392,   28096447250432,   4444945756416;
  9: 1207269217792, 43461691840512, 391155226564608, 1622569828712448, 3802898036044800, 5476173171904512, 4969120100431872, 2781548277792768, 880099259770368, 120726921779200;

The row polynomials satisfy (1+(3*n-1)*x+n^2*x^2)*P(n,x) = (22+(66*n+66)*x+(22*n^2+110*n+66)*x^2+(22*n^2+44*n+22)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((22 + (66*n + 66)*x + (22*n^2 + 110*n + 66)*x^2 + (22*n^2 + 44*n + 22)*x^3)*P[n - 1, x])/(1 + (3*n - 1)*x + n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((22 + (66*n + 66)*x + (22*n^2 + 110*n + 66)*x^2 + (22*n^2 + 44*n + 22)*x^3)*v[i - 1])/(1 + (3*n - 1)*x + n^2*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 basis_map_A013609 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^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
        • 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 read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Iterated real-rootedness preservers (above)
            • Parent-row provers of polytool polytool ? step › direct check
        • ↑ Parent-row provers of polytool (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.?
        • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A026729 → basis A007318 → (θ+1)²
• A178821 → (θ+1)²
• A038208 → D → (θ+1)²
• A156992 → (θ+1)²
• A125092 → reverse → D
• A125092 → reverse
• A344565 → (θ+1)²
• A156992 → basis A013609 → (θ+1)²
• A085880 → (θ+1)²
• basis-map coefficients of A113214, shifted → basis A013609 → (θ+1)²
• A356546 → (θ+1)²
• A003506 → D → (θ+1)²
• A027465 → basis A013611 → (θ+1)²
• A085881 → basis A013609 → (θ+1)²
• A038325 → (θ+1)²
• A178820 → basis A013609 → (θ+1)²
• A134346 → (θ+1)²
• A135838 → (θ+1)²
• A038208 → (θ+1)²
• A038330 → basis A013611 → (θ+1)²
• A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
• A356546 → basis A013609 → (θ+1)²
• basis-map coefficients of A127647 → (θ+1)²
• A028326 → basis A013609 → (θ+1)²
• A038221 → basis A013609 → (θ+1)²
• A085881 → (θ+1)²
• A124860 → (θ+1)²
• A196347 → (θ+1)²
• A038338 → basis A013609 → (θ+1)²
• A196347 → basis A013609 → (θ+1)²
• basis-map coefficients of A112227 → x ↦ x+1 → (θ+1)²
• A038338 → (θ+1)²
• A120910 → basis A013610 → (θ+1)²
• A085880 → D → (θ+1)²
• basis-map coefficients of A026729, shifted → basis A007318 → (θ+1)²
• A178821 → basis A013609 → (θ+1)²
• basis-map coefficients of A113214, shifted → D → (θ+1)²
• A178822 → (θ+1)²
• A028326 → D → (θ+1)²
• A376827 → (θ+1)²
• A085880 → basis A013609 → (θ+1)²
• A178822 → basis A013609 → (θ+1)²
• basis-map coefficients of A113214, shifted → (θ+1)²
• A028326 → (θ+1)²
• A003506 → (θ+1)²
• A135065 → (θ+1)²
• A038221 → (θ+1)²
• A131431 → x ↦ x+1 → (θ+1)²
• A178820 → (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.