A038319 → basis A013609 → D

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Submission text ?

Triangle begins:
  0:             320,             200;
  1:            7680,            9600,             3000;
  2:          163840,          307200,           192000,            40000;
  3:         3276800,         8192000,          7680000,          3200000,           500000;
  4:        62914560,       196608000,        245760000,        153600000,         48000000,          6000000;
  5:      1174405120,      4404019200,       6881280000,       5734400000,       2688000000,        672000000,         70000000;
  6:     21474836480,     93952409600,     176160768000,     183500800000,     114688000000,      43008000000,       8960000000,       800000000;
  7:    386547056640,   1932735283200,    4227858432000,    5284823040000,    4128768000000,    2064384000000,     645120000000,    115200000000,      9000000000;
  8:   6871947673600,  38654705664000,   96636764160000,  140928614400000,  132120576000000,   82575360000000,   34406400000000,   9216000000000,   1440000000000,   100000000000;
  9: 120946279055360, 755914244096000, 2126008811520000, 3543348019200000, 3875536896000000, 2906652672000000, 1513881600000000, 540672000000000, 126720000000000, 17600000000000, 1100000000000;

The row polynomials satisfy (n+1)*P(n,x) = (16*n+32+(10*n+20)*x)*P(n-1,x) for n >= 1, with P(0,x) = 320 + 200*x.

P[0, x_] := 320 + 200*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 32 + (10*n + 20)*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] = 320 + 200*x; for(n = 1, N, my(i = n + 1); v[i] = ((16*n + 32 + (10*n + 20)*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_A013609 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)
    • Geometric basis map (elementary) polytool ? step ›

      • from basis_map_A013609 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 (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.?
        • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A013622 → basis A007318
• A038329 → basis A013610 → x ↦ x/(x+1)
• basis-map coefficients of A112227 → basis A013611 → basis A013609
• A013622 → basis A007318 → D
• A013611 → D → basis A013609
• A120910 → basis A013609 → basis A013611
• A038220 → basis A013611
• basis-map coefficients of A026729 → basis A013611 → basis A013609
• A038220 → basis A013611 → D
• A038319 → basis A013609
• A038319 → basis A013609 → reverse
• A038259 → basis A007318 → basis A013609
• A130123 → basis A013611 → basis A013609
• A136572 → basis A013611 → basis A013609
• A038336 → x ↦ x+1 → basis A013609
• A038334 → basis A013611
• A038271 → basis A013611 → basis A007318

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.