A038319 → basis A013610 → D

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Submission text ?

Triangle begins:
  0:             480,              450;
  1:           11520,            21600,            10125;
  2:          245760,           691200,           648000,            202500;
  3:         4915200,         18432000,         25920000,          16200000,           3796875;
  4:        94371840,        442368000,        829440000,         777600000,         364500000,          68343750;
  5:      1761607680,       9909043200,      23224320000,       29030400000,       20412000000,        7654500000,        1196015625;
  6:     32212254720,     211392921600,     594542592000,      928972800000,      870912000000,      489888000000,      153090000000,       20503125000;
  7:    579820584960,    4348654387200,   14269022208000,    26754416640000,    31352832000000,    23514624000000,    11022480000000,     2952450000000,     345990234375;
  8:  10307921510400,   86973087744000,  326149079040000,   713451110400000,  1003290624000000,   940584960000000,   587865600000000,   236196000000000,   55358437500000,    5766503906250;
  9: 181419418583040, 1700807049216000, 7175279738880000, 17938199347200000, 29429858304000000, 33108590592000000, 25866086400000000, 13856832000000000, 4871542500000000, 1014904687500000, 95147314453125;

The row polynomials satisfy (n+1)*P(n,x) = (16*n+32+(15*n+30)*x)*P(n-1,x) for n >= 1, with P(0,x) = 480 + 450*x.

P[0, x_] := 480 + 450*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 32 + (15*n + 30)*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] = 480 + 450*x; for(n = 1, N, my(i = n + 1); v[i] = ((16*n + 32 + (15*n + 30)*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)*11^(i-j)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^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
        • Geometric basis map (elementary) polytool ? step ›

          • from basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.?
            • Geometric basis map (elementary) polytool ? step ›

              • from Triangle of coefficients in expansion of (1+5x)^n.?
                • 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 of coefficients in expansion of (1+5x)^n.?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.?
            • ↑ Geometric basis map (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A027467 → x ↦ x/(x+1)
• A038259 → basis A007318 → basis A013610
• A038319 → basis A013610
• A038319 → basis A013610 → reverse
• A038336 → x ↦ x+1 → basis A013610

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.