A038319 → D+1

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Submission text ?

Triangle begins:
  0:           231,           160,             25;
  1:          3146,          3465,           1200,            125;
  2:         41261,         62920,          34650,           8000,            625;
  3:        527076,       1031525,         786500,         288750,          50000,          3125;
  4:       6603091,      15812280,       15472875,        7865000,        2165625,        300000,         15625;
  5:      81491806,     231108185,      276714900,      180516875,       68818750,      15159375,       1750000,         78125;
  6:     993845721,    3259672240,     4622163700,     3689532000,     1805168750,     550550000,     101062500,      10000000,       390625;
  7:   12004097336,   44723057445,    73342625400,    69332455500,    41507235000,   16246518750,    4129125000,     649687500,     56250000,     1953125;
  8:  143834809151,  600204866800,  1118076436125,  1222377090000,   866655693750,  415072350000,  135387656250,   29493750000,   4060546875,   312500000,    9765625;
  9: 1711870023666, 7910914503305, 16505633837000, 20498067995625, 16807684987500, 9533212631250, 3804829875000, 1063760156250, 202769531250, 24814453125, 1718750000, 48828125;

The row polynomials satisfy (5*n+16+5*x)*P(n,x) = (55*n+231+(25*n+160)*x+25*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 231 + 160*x + 25*x^2.

P[0, x_] := 231 + 160*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((55*n + 231 + (25*n + 160)*x + 25*x^2)*P[n - 1, x])/(5*n + 16 + 5*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 231 + 160*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((55*n + 231 + (25*n + 160)*x + 25*x^2)*v[i - 1])/(5*n + 16 + 5*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 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 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 ?

• A038336 → x ↦ x+1 → D+1
• A038259 → basis A007318 → D+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.