A038329 → basis A013609 → D

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Submission text ?

Triangle begins:
  0:            180,              72;
  1:           4050,            3240,             648;
  2:          81000,           97200,           38880,            5184;
  3:        1518750,         2430000,         1458000,          388800,           38880;
  4:       27337500,        54675000,        43740000,        17496000,         3499200,         279936;
  5:      478406250,      1148175000,      1148175000,       612360000,       183708000,       29393280,        1959552;
  6:     8201250000,     22963500000,     27556200000,     18370800000,      7348320000,     1763596800,      235146240,      13436928;
  7:   138396093750,    442867500000,    620014500000,    496011600000,    248005800000,    79361856000,    15872371200,    1813985280,      90699264;
  8:  2306601562500,   8303765625000,  13286025000000,  12400290000000,   7440174000000,  2976069600000,   793618560000,  136048896000,   13604889600,   604661760;
  9: 38058925781250, 152235703125000, 274024265625000, 292292550000000, 204604785000000, 98210296800000, 32736765600000, 7482689280000, 1122403392000, 99769190400, 3990767616;

The row polynomials satisfy (n+1)*P(n,x) = (15*n+30+(6*n+12)*x)*P(n-1,x) for n >= 1, with P(0,x) = 180 + 72*x.

P[0, x_] := 180 + 72*x;
P[n_, x_] := P[n, x] = Expand[Together[((15*n + 30 + (6*n + 12)*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 + 72*x; for(n = 1, N, my(i = n + 1); v[i] = ((15*n + 30 + (6*n + 12)*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)*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) 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.?
            • 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
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

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