A139548 → x ↦ x+1 → D+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:          1;
  1:         16,          12,           2;
  2:        192,         320,         192,           48,            4;
  3:       2048,        5376,        5760,         3200,          960,          144,            8;
  4:      20480,       73728,      114688,       100352,        53760,        17920,         3584,          384,           16;
  5:     196608,      901120,     1843200,      2211840,      1720320,       903168,       322560,        76800,        11520,         960,          32;
  6:    1835008,    10223616,    25952256,     39649280,     40550400,     29196288,     15138816,      5677056,      1520640,      281600,       33792,       2304,         64;
  7:   16777216,   110100480,   333971456,    620232704,    787218432,    721616896,    492011520,    253034496,     98402304,    28700672,     6150144,     931840,      93184,      5376,      128;
  8:  150994944,  1140850688,  4026531840,   8808038400,  13358858240,  14885584896,  12595494912,   8247050240,   4217241600,  1686896640,   524812288,  125239296,   22364160,   2867200,   245760,   12288,    256;
  9: 1342177280, 11475615744, 46204452864, 116366770176, 205353123840, 269525975040, 272520708096, 216904237056, 137650765824, 70099927040, 28677242880, 9385279488, 2433220608, 491323392, 75202560, 8355840, 626688, 27648, 512;

The row polynomials satisfy (2*n+x)*P(n,x) = (16*n+16+(16*n+24)*x+(4*n+12)*x^2+2*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 16 + (16*n + 24)*x + (4*n + 12)*x^2 + 2*x^3)*P[n - 1, x])/(2*n + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n + 16 + (16*n + 24)*x + (4*n + 12)*x^2 + 2*x^3)*v[i - 1])/(2*n + 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

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A007318 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
            • 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
            • 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 (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

• A034870 → x ↦ x+1 → D+1

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.