A013612 → basis A013610 → D

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Submission text ?

Triangle begins:
  0:        180,          450;
  1:       1620,         8100,         10125;
  2:      12960,        97200,        243000,         202500;
  3:      97200,       972000,       3645000,        6075000,        3796875;
  4:     699840,      8748000,      43740000,      109350000,      136687500,        68343750;
  5:    4898880,     73483200,     459270000,     1530900000,     2870437500,      2870437500,      1196015625;
  6:   33592320,    587865600,    4408992000,    18370800000,    45927000000,     68890500000,     57408750000,     20503125000;
  7:  226748160,   4534963200,   39680928000,   198404640000,   620014500000,   1240029000000,   1550036250000,   1107168750000,    345990234375;
  8: 1511654400,  34012224000,  340122240000,  1984046400000,  7440174000000,  18600435000000,  31000725000000,  33215062500000,  20759414062500,   5766503906250;
  9: 9976919040, 249422976000, 2806008480000, 18706723200000, 81841914000000, 245525742000000, 511511962500000, 730731375000000, 685060664062500, 380589257812500, 95147314453125;

The row polynomials satisfy (n+1)*P(n,x) = (6*n+12+(15*n+30)*x)*P(n-1,x) for n >= 1, with P(0,x) = 180 + 450*x.

P[0, x_] := 180 + 450*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 12 + (15*n + 30)*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] = 180 + 450*x; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 12 + (15*n + 30)*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_A013610 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 basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^n.?
            • ↑ Geometric basis map (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^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
            • 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

Related ?

Same fingerprint ?

• A038329 → basis A013609 → D
• A038291 → basis A013611 → reverse
• A013622 → basis A013611
• A038334 → x ↦ x+1
• A120910 → basis A013609 → basis A007318
• A178820 → basis A013610 → x ↦ x/(x+1)
• A038220 → basis A007318 → reverse
• A156992 → basis A013610 → x ↦ x/(x+1)
• A013622 → basis A013611 → reverse
• A038208 → basis A013610 → x ↦ x/(x+1)
• A178821 → basis A013610 → x ↦ x/(x+1)
• A013620 → x ↦ x/(x+1)
• A038334 → x ↦ x+1 → reverse
• A038220 → basis A007318
• A038329 → basis A013609
• A038220 → basis A007318 → D
• A038338 → basis A013610 → x ↦ x/(x+1)
• A038291 → basis A013611 → D
• A038327 → basis A013609 → basis A013610
• A038221 → basis A013610 → x ↦ x/(x+1)
• A178822 → basis A013610 → x ↦ x/(x+1)
• A038291 → basis A013611
• A085880 → basis A013610 → x ↦ x/(x+1)
• A196347 → basis A013610 → x ↦ x/(x+1)
• A028326 → basis A013610 → x ↦ x/(x+1)
• A013622 → basis A013611 → D
• A038231 → D → basis A013609
• A013612 → basis A013610
• A038334 → x ↦ x+1 → D

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.