A013612 → basis A013609 → D

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Submission text ?

Triangle begins:
  0:        120,          200;
  1:       1080,         3600,         3000;
  2:       8640,        43200,        72000,         40000;
  3:      64800,       432000,      1080000,       1200000,         500000;
  4:     466560,      3888000,     12960000,      21600000,       18000000,        6000000;
  5:    3265920,     32659200,    136080000,     302400000,      378000000,      252000000,       70000000;
  6:   22394880,    261273600,   1306368000,    3628800000,     6048000000,     6048000000,     3360000000,      800000000;
  7:  151165440,   2015539200,  11757312000,   39191040000,    81648000000,   108864000000,    90720000000,    43200000000,     9000000000;
  8: 1007769600,  15116544000, 100776960000,  391910400000,   979776000000,  1632960000000,  1814400000000,  1296000000000,   540000000000,  100000000000;
  9: 6651279360, 110854656000, 831409920000, 3695155200000, 10777536000000, 21555072000000, 29937600000000, 28512000000000, 17820000000000, 6600000000000, 1100000000000;

The row polynomials satisfy (n+1)*P(n,x) = (6*n+12+(10*n+20)*x)*P(n-1,x) for n >= 1, with P(0,x) = 120 + 200*x.

P[0, x_] := 120 + 200*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 12 + (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] = 120 + 200*x; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 12 + (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 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

Related ?

Same fingerprint ?

• A120910 → basis A013609 → x ↦ x/(x+1)
• A013622: Triangle of coefficients in expansion of (3+5x)^n.
• A038329 → basis A013610 → reverse
• A178821 → basis A013610 → x ↦ x+1
• A038284 → basis A007318 → basis A013609
• A178822 → basis A013610 → basis A007318
• A038319 → basis A013611 → basis A013610
• A038335 → x ↦ x+1 → basis A013609
• A038231 → D → basis A013610
• A013612 → basis A013609
• A038271 → basis A013611
• A038329 → basis A013610 → D
• A013612 → basis A013609 → reverse
• A038255 → basis A013610 → basis A013609
• A038329 → basis A013610
• A027467 → basis A013611 → basis A013610
• A038221 → basis A013610 → basis A007318
• A013620 → basis A007318 → D
• A013620 → basis A007318
• A028326 → basis A013610 → x ↦ x+1
• A196347 → basis A013610 → basis A007318
• A013622 → D
• A178820 → basis A013610 → x ↦ x+1
• A038271 → basis A013611 → D
• A038334 → x ↦ x/(x+1)
• A038338 → basis A013610 → x ↦ x+1
• A156992 → basis A013610 → x ↦ x+1
• A038243 → basis A013611 → x ↦ x/(x+1)
• A013622 → basis A013609 → basis A013610
• A085880 → basis A013610 → x ↦ x+1
• A038271 → basis A013610 → 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.