A013612 → x ↦ x+1 → D

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Submission text ?

Triangle begins:
  0:         60,          50;
  1:        540,         900,          375;
  2:       4320,       10800,         9000,         2500;
  3:      32400,      108000,       135000,        75000,        15625;
  4:     233280,      972000,      1620000,      1350000,       562500,        93750;
  5:    1632960,     8164800,     17010000,     18900000,     11812500,      3937500,       546875;
  6:   11197440,    65318400,    163296000,    226800000,    189000000,     94500000,     26250000,      3125000;
  7:   75582720,   503884800,   1469664000,   2449440000,   2551500000,   1701000000,    708750000,    168750000,    17578125;
  8:  503884800,  3779136000,  12597120000,  24494400000,  30618000000,  25515000000,  14175000000,   5062500000,  1054687500,   97656250;
  9: 3325639680, 27713664000, 103926240000, 230947200000, 336798000000, 336798000000, 233887500000, 111375000000, 34804687500, 6445312500, 537109375;

The row polynomials satisfy (n+1)*P(n,x) = (6*n+12+(5*n+10)*x)*P(n-1,x) for n >= 1, with P(0,x) = 60 + 50*x.

P[0, x_] := 60 + 50*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 12 + (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] = 60 + 50*x; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 12 + (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 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)

Related ?

Same fingerprint ?

• A156992 → basis A013610 → basis A013609
• A147716 → basis A013610 → x ↦ x/(x+1)
• A038334 → basis A013610
• A038336 → D
• A038329 → x ↦ x+1 → x ↦ x/(x+1)
• A038259: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.
• A038255 → basis A013610 → basis A013611
• A038338 → basis A013610 → basis A013609
• A178822 → basis A013610 → basis A013609
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
• A038271 → basis A013609
• A085880 → basis A013610 → basis A013609
• A178820 → basis A013610 → basis A013609
• A038220 → basis A013610 → D
• A038259 → D
• A013612 → x ↦ x+1
• A013612 → x ↦ x+1 → reverse
• A196347 → basis A013610 → basis A013609
• A178821 → basis A013610 → basis A013609
• A120910 → basis A013609 → basis A013610
• A038220 → basis A013610
• A028326 → basis A013610 → basis A013609
• A038271 → basis A013609 → D
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
• A038334 → basis A013610 → D
• A038314 → D
• A038291 → basis A013609 → x ↦ x/(x+1)
• A038221 → basis A013610 → basis A013609
• A038334 → basis A013610 → reverse

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.