A038226 → basis A013611 → D+1

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Submission text ?

Triangle begins:
  0:           825,            2752,             1024;
  1:         12947,           79200,           132096,             32768;
  2:        185009,         1657216,          5068800,           5636096,            1048576;
  3:       2503611,        29601440,        132577280,         270336000,          225443840,           33554432;
  4:      32693353,       480693312,       2841738240,        8484945920,        12976128000,         8657043456,          1073741824;
  5:     416316835,      7323311072,      53837650944,      212183121920,       475156971520,       581330534400,        323196289024,         34359738368;
  6:    5203074657,    106577109760,     937383817216,     4594146213888,     13579719802880,     24328036941824,      24803436134400,      11819749998592,       1099511627776;
  7:   64093305419,   1498485501216,   15347103805440,    89988846452736,    330778527399936,    782191860645888,    1167745773207552,    1020484229529600,     425510999949312,      35184372088832;
  8:  780480685721,  20509857734080,  239757680194560,  1637024405913600,   7199107716218880,  21169825753595904,   41716899234447360,   53382663918059520,   40819369181184000,   15129279998197760,   1125899906842624;
  9: 9415285130163, 274729201373792, 3609734961198080, 28131567809495040, 144058147720396800, 506817183221809152, 1241963110877626368, 2097764075789352960, 2348837212394618880, 1596490883530752000, 532550655936561152, 36028797018963968;

The row polynomials satisfy (32*n+43+32*x)*P(n,x) = (352*n+825+(1024*n+2752)*x+1024*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 825 + 2752*x + 1024*x^2.

P[0, x_] := 825 + 2752*x + 1024*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((352*n + 825 + (1024*n + 2752)*x + 1024*x^2)*P[n - 1, x])/(32*n + 43 + 32*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 825 + 2752*x + 1024*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((352*n + 825 + (1024*n + 2752)*x + 1024*x^2)*v[i - 1])/(32*n + 43 + 32*x)); 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_A013611 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*8^(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
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (above)
                • 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
            • 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
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

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.