A114655 → x ↦ x+1 → D+1

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Submission text ?

Triangle begins:
   1:       2;
   2:       8,        2;
   3:      38,       20,        2;
   4:     192,      162,       36,        2;
   5:    1002,     1192,      436,       56,       2;
   6:    5336,     8290,     4360,      940,      80,      2;
   7:   28814,    55596,    38790,    12200,    1770,    108,      2;
   8:  157184,   363650,   319116,   134750,   28840,   3038,    140,    2;
   9:  864146,  2336144,  2483432,  1332240,  386428,  60592,   4872,  176,   2;
  10: 4780008, 14806530, 18547920, 12155640, 4515840, 966924, 116592, 7416, 216, 2;

The row polynomials satisfy (n^3+1+(2*n^2-2*n-4)*x+(n+1)*x^2)*P(n,x) = (5*n^3-3*n+2+(3*n^3+6*n^2-22*n+5)*x+(6*n^2-10*n+8)*x^2+(3*n-3)*x^3)*P(n-1,x) + (2*n^2+n-1+(4*n-4)*x-(2*n^2+n-3)*x^2-(4*n-4)*x^3-2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.

P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((5*n^3 - 3*n + 2 + (3*n^3 + 6*n^2 - 22*n + 5)*x + (6*n^2 - 10*n + 8)*x^2 + (3*n - 3)*x^3)*P[n - 1, x] + (2*n^2 + n - 1 + (4*n - 4)*x - (2*n^2 + n - 3)*x^2 - (4*n - 4)*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 + 1 + (2*n^2 - 2*n - 4)*x + (n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((5*n^3 - 3*n + 2 + (3*n^3 + 6*n^2 - 22*n + 5)*x + (6*n^2 - 10*n + 8)*x^2 + (3*n - 3)*x^3)*v[i - 1] + (2*n^2 + n - 1 + (4*n - 4)*x - (2*n^2 + n - 3)*x^2 - (4*n - 4)*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^3 + 1 + (2*n^2 - 2*n - 4)*x + (n + 1)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 shift_x_plus_one of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascen…?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schr…?
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k…?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schr…?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schr…?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schr…?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A114656 → basis A007318 → D+1

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.