A038319 → D

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Submission text ?

Triangle begins:
  0:           110,            50;
  1:          1815,          1650,            375;
  2:         26620,         36300,          16500,           2500;
  3:        366025,        665500,         453750,         137500,          15625;
  4:       4831530,      10980750,        9982500,        4537500,        1031250,         93750;
  5:      62004635,     169103550,      192163125,      116462500,       39703125,       7218750,        546875;
  6:     779486840,    2480185400,     3382071000,     2562175000,     1164625000,     317625000,      48125000,      3125000;
  7:    9646149645,   35076907800,    55804171500,    50731065000,    28824468750,   10481625000,    2382187500,    309375000,     17578125;
  8:  117897384550,  482307482250,   876922695000,   930069525000,   634138312500,  288244687500,   87346875000,  17015625000,   1933593750,    97656250;
  9: 1426558353055, 6484356150250, 13263455761875, 16076916075000, 12788455968750, 6975521437500, 2642242968750, 686296875000, 116982421875, 11816406250, 537109375;

The row polynomials satisfy (n+1)*P(n,x) = (11*n+22+(5*n+10)*x)*P(n-1,x) for n >= 1, with P(0,x) = 110 + 50*x.

P[0, x_] := 110 + 50*x;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 22 + (5*n + 10)*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] = 110 + 50*x; for(n = 1, N, my(i = n + 1); v[i] = ((11*n + 22 + (5*n + 10)*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 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 ?

• A038301 → basis A013611
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.
• A038220 → basis A013610 → x ↦ x/(x+1)
• A038336 → x ↦ x+1 → reverse
• A038336 → x ↦ x+1
• A038314 → x ↦ x/(x+1)
• A038301 → basis A013611 → reverse
• A038259 → basis A007318 → reverse
• A038301 → basis A013611 → D
• A038259 → basis A007318 → D
• A038336 → x ↦ x+1 → D
• A038259 → basis A007318
• A038319 → reverse
• A013618 → basis A013610 → basis A013610
• A038271 → basis A013609 → x ↦ x+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.