A038284 → basis A013610 → D

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Submission text ?

Triangle begins:
  0:            504,             648;
  1:          10584,           27216,            17496;
  2:         197568,          762048,           979776,            419904;
  3:        3457440,        17781120,         34292160,          29393280,           9447840;
  4:       58084992,       373403520,        960180480,        1234517760,         793618560,         204073344;
  5:      948721536,      7318708992,      23524421760,       40327580160,       38887309440,       19999187712,        4285540224;
  6:    15179544576,    136615901184,     526947047424,     1129172244480,     1451792885760,     1119954511872,      479980505088,       88159684608;
  7:   239077827072,   2459086221312,   11065887995904,    28455140560896,    45731475901440,    47038089498624,    30238771820544,    11108120260608,     1785233613312;
  8:  3718988421120,  43034008872960,  221317759918080,   663953279754240,  1280481325240320,  1646333132451840,  1411142684958720,   777568418242560,   249932705863680,   35704672266240;
  9: 57272421685248, 736359707381760, 4260366878423040, 14606972154593280, 32865687347834880, 50707060479516672, 54328993370910720, 39915178803118080, 19244818351503360, 5498519529000960, 706952510871552;

The row polynomials satisfy (n+1)*P(n,x) = (14*n+28+(18*n+36)*x)*P(n-1,x) for n >= 1, with P(0,x) = 504 + 648*x.

P[0, x_] := 504 + 648*x;
P[n_, x_] := P[n, x] = Expand[Together[((14*n + 28 + (18*n + 36)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 504 + 648*x; for(n = 1, N, my(i = n + 1); v[i] = ((14*n + 28 + (18*n + 36)*x)*v[i - 1])/(n + 1)); 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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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 basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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 basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
            • ↑ Geometric basis map (elementary) (above)
            • 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)*2^j.?
                • Geometric basis map (elementary) polytool ? step ›

                  • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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) (above)

Related ?

Same fingerprint ?

• A038330 → basis A013610 → basis A013610
• A013610 → basis A007318 → basis A013610
• A038321 → basis A013609
• A038255 → basis A013609 → x ↦ x/(x+1)
• A038335 → basis A013610 → D
• A038335 → basis A013610 → reverse
• A038335 → basis A013610
• A013620 → basis A013610 → basis A013609
• A038284 → basis A013610
• A038321 → basis A013609 → D

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.