A038329 → basis A013609 → (θ+1)²

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Submission text ?

Triangle begins:
  0:           225,             720,             324;
  1:          3375,           16200,           14580,             3456;
  2:         50625,          324000,          437400,           207360,            32400;
  3:        759375,         6075000,        10935000,          7776000,          2430000,           279936;
  4:      11390625,       109350000,       246037500,        233280000,        109350000,         25194240,         2286144;
  5:     170859375,      1913625000,      5166787500,       6123600000,       3827250000,       1322697600,       240045120,        17915904;
  6:    2562890625,     32805000000,    103335750000,     146966400000,     114817500000,      52907904000,     14402707200,      2149908480,      136048896;
  7:   38443359375,    553584375000,   1992903750000,    3306744000000,    3100072500000,    1785641760000,    648121824000,    145118822400,    18366600960,     1007769600;
  8:  576650390625,   9226406250000,  37366945312500,   70858800000000,   77501812500000,   53569252800000,  24304568400000,   7255941120000,  1377495072000,   151165440000,    7316407296;
  9: 8649755859375, 152235703125000, 685060664062500, 1461462750000000, 1826828437500000, 1473154452000000, 802050757200000, 299307571200000, 75762228960000, 12471148800000, 1207207203840, 52242776064;

The row polynomials satisfy (25+(30*n+50)*x+(4*n^2+16*n+16)*x^2)*P(n,x) = (375+(450*n+1350)*x+(60*n^2+540*n+1020)*x^2+(24*n^2+144*n+216)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 720*x + 324*x^2.

P[0, x_] := 225 + 720*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((375 + (450*n + 1350)*x + (60*n^2 + 540*n + 1020)*x^2 + (24*n^2 + 144*n + 216)*x^3)*P[n - 1, x])/(25 + (30*n + 50)*x + (4*n^2 + 16*n + 16)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 225 + 720*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((375 + (450*n + 1350)*x + (60*n^2 + 540*n + 1020)*x^2 + (24*n^2 + 144*n + 216)*x^3)*v[i - 1])/(25 + (30*n + 50)*x + (4*n^2 + 16*n + 16)*x^2)); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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 → (θ+1)²
• A038334 → x ↦ x+1 → (θ+1)²
• A038291 → basis A013611 → (θ+1)²

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.