A038338 → basis A013610 → θ+1

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Submission text ?

Triangle begins:
  0:              576,              3456,               3888;
  1:            13824,            124416,             279936,              186624;
  2:           331776,           3981312,           13436928,            17915904,              8398080;
  3:          7962624,         119439360,          537477120,          1074954240,           1007769600,            362797056;
  4:        191102976,        3439853568,        19349176320,         51597803520,          72559411200,          52242776064,          15237476352;
  5:       4586471424,       96315899904,       650132324352,       2167107747840,        4063327027200,        4388393189376,        2559896027136,         626913312768;
  6:     110075314176,     2641807540224,     20804234379264,      83216937517056,      195039697305600,      280857164120064,      245750018605056,      120367356051456,       25389989167104;
  7:    2641807540224,    71328803586048,    641959232274432,    2995809750614016,     8425714923601920,    15166286862483456,    17694001339564032,    12999674453557248,     5484237660094464,     1015599566684160;
  8:   63403380965376,  1902101428961280,  19258776968232960,  102713477163909120,   337028596944076800,   727981769399205888,  1061640080373841920,  1039973956284579840,   658108519211335680,   243743896004198400,    40217742840692736;
  9: 1521681143169024, 50215477724577792, 564924124401500160, 3389544746409000960, 12710792799033753600, 32031197853565059072, 56054596243738853376, 68638281114782269440, 57913549690597539840, 32174194272554188800, 10617484109942882304, 1579460446107205632;

The row polynomials satisfy (2+(3*n+6)*x)*P(n,x) = (48+(72*n+288)*x+(108*n+324)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 3456*x + 3888*x^2.

P[0, x_] := 576 + 3456*x + 3888*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((48 + (72*n + 288)*x + (108*n + 324)*x^2)*P[n - 1, x])/(2 + (3*n + 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 576 + 3456*x + 3888*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((48 + (72*n + 288)*x + (108*n + 324)*x^2)*v[i - 1])/(2 + (3*n + 6)*x)); 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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
    • 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)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
        • 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) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • 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)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A013620 → θ+1
• A178820 → basis A013610 → θ+1
• A156992 → basis A013610 → θ+1
• A028326 → basis A013610 → θ+1
• A178821 → basis A013610 → θ+1
• A178822 → basis A013610 → θ+1
• A196347 → basis A013610 → θ+1
• A038221 → basis A013610 → θ+1
• A085880 → basis A013610 → θ+1
• A038208 → basis A013610 → θ+1

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.