A038338 → basis A013609 → (θ+1)²

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Triangle begins:
  0:              576,              4608,               5184;
  1:            13824,            165888,             373248,              221184;
  2:           331776,           5308416,           17915904,            21233664,              8294400;
  3:          7962624,         159252480,          716636160,          1274019840,            995328000,            286654464;
  4:        191102976,        4586471424,        25798901760,         61152952320,          71663616000,          41278242816,           9364045824;
  5:       4586471424,      128421199872,       866843099136,       2568423997440,        4013162496000,        3467372396544,        1573159698432,         293534171136;
  6:     110075314176,     3522410053632,     27738979172352,      98627481501696,      192631799808000,      221911833378816,      151023331049472,       56358560858112,        8916100448256;
  7:    2641807540224,    95105071448064,    855945643032576,    3550589334061056,     8321693751705600,    11983239002456064,    10873679835561984,     6086724572676096,     1925877696823296,     264180754022400;
  8:   63403380965376,  2536135238615040,  25678369290977280,  121734491453521920,   332867750068224000,   575195472117891072,   652420790133719040,   486937965814087680,   231105323618795520,   63403380965376000,    7671809096810496;
  9: 1521681143169024, 66953970299437056, 753232165868666880, 4017238217966223360, 12553869431144448000, 25308600773187207168, 34447817719060365312, 32137905743729786880, 20337268478454005760, 8369246287429632000, 2025357601557970944, 219122084616339456;

The row polynomials satisfy (1+(3*n+5)*x+(n^2+4*n+4)*x^2)*P(n,x) = (24+(72*n+216)*x+(24*n^2+216*n+408)*x^2+(24*n^2+144*n+216)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 4608*x + 5184*x^2.

P[0, x_] := 576 + 4608*x + 5184*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((24 + (72*n + 216)*x + (24*n^2 + 216*n + 408)*x^2 + (24*n^2 + 144*n + 216)*x^3)*P[n - 1, x])/(1 + (3*n + 5)*x + (n^2 + 4*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 576 + 4608*x + 5184*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24 + (72*n + 216)*x + (24*n^2 + 216*n + 408)*x^2 + (24*n^2 + 144*n + 216)*x^3)*v[i - 1])/(1 + (3*n + 5)*x + (n^2 + 4*n + 4)*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 whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
        • ↑ Geometric basis map (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^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
            • 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)*12^j.?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (above)
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A003506 → D → (θ+1)²
• A038221 → (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).
• A156992 → basis A013609 → (θ+1)²
• A085880 → D → (θ+1)²
• A196347 → (θ+1)²
• A038325 → (θ+1)²
• A085880 → basis A013609 → (θ+1)²
• basis-map coefficients of A113214, shifted → (θ+1)²
• A178821 → (θ+1)²
• basis-map coefficients of A113214, shifted → D → (θ+1)²
• A178822 → basis A013609 → (θ+1)²
• basis-map coefficients of A113214, shifted → basis A013609 → (θ+1)²
• A376827 → (θ+1)²
• A135838 → (θ+1)²
• A356546 → basis A013609 → (θ+1)²
• A134346 → (θ+1)²
• A027465 → basis A013611 → (θ+1)²
• A028326 → D → (θ+1)²
• A085880 → (θ+1)²
• A356546 → (θ+1)²
• A135065 → (θ+1)²
• A085881 → (θ+1)²
• A124860 → (θ+1)²
• A120910 → basis A013610 → (θ+1)²
• basis-map coefficients of A026729, shifted → basis A007318 → (θ+1)²
• basis-map coefficients of A112227 → x ↦ x+1 → (θ+1)²
• A038325 → basis A013609 → (θ+1)²
• A344565 → (θ+1)²
• A038208 → (θ+1)²
• A178820 → basis A013609 → (θ+1)²
• A085881 → basis A013609 → (θ+1)²
• basis-map coefficients of A026729 → basis A007318 → (θ+1)²
• basis-map coefficients of A127647 → (θ+1)²
• A178822 → (θ+1)²
• A038221 → basis A013609 → (θ+1)²
• A178821 → basis A013609 → (θ+1)²
• A038208 → D → (θ+1)²
• A028326 → (θ+1)²
• A028326 → basis A013609 → (θ+1)²
• A156992 → (θ+1)²
• A003506 → (θ+1)²
• A038338 → (θ+1)²
• A131431 → x ↦ x+1 → (θ+1)²
• A125092 → reverse → D
• A196347 → basis A013609 → (θ+1)²
• A125092 → reverse
• 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.