A114655 → reverse → D+1

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Triangle begins:
   1:   2;
   2:   6,    4;
   3:  14,   28,      8;
   4:  26,  120,     96,     16;
   5:  42,  360,    640,    288,      32;
   6:  62,  860,   2800,   2720,     800,      64;
   7:  86, 1764,   9240,  16240,   10080,    2112,     128;
   8: 114, 3248,  25088,  70560,   78400,   34048,    5376,    256;
   9: 146, 5520,  59136, 244608,  432768,  333312,  107520,  13312,   512;
  10: 182, 8820, 125280, 717696, 1862784, 2274048, 1290240, 322560, 32256, 1024;

The row polynomials satisfy (2*n^3-4*n^2+6+(4*n^2-5*n-9)*x+(2*n+2)*x^2)*P(n,x) = (2*n^3+2+(12*n^3-36*n^2+37*n-23)*x+(24*n^2-64*n+44)*x^2+(12*n-12)*x^3)*P(n-1,x) + ((4*n^2-6*n+4)*x-(8*n^2-20*n+18)*x^2-(16*n-24)*x^3-8*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[((2*n^3 + 2 + (12*n^3 - 36*n^2 + 37*n - 23)*x + (24*n^2 - 64*n + 44)*x^2 + (12*n - 12)*x^3)*P[n - 1, x] + ((4*n^2 - 6*n + 4)*x - (8*n^2 - 20*n + 18)*x^2 - (16*n - 24)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(2*n^3 - 4*n^2 + 6 + (4*n^2 - 5*n - 9)*x + (2*n + 2)*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] = ((2*n^3 + 2 + (12*n^3 - 36*n^2 + 37*n - 23)*x + (24*n^2 - 64*n + 44)*x^2 + (12*n - 12)*x^3)*v[i - 1] + ((4*n^2 - 6*n + 4)*x - (8*n^2 - 20*n + 18)*x^2 - (16*n - 24)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(2*n^3 - 4*n^2 + 6 + (4*n^2 - 5*n - 9)*x + (2*n + 2)*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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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 [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) (above)

Related ?

Same fingerprint ?

• A114656 → reverse → D+1
• A114687 → 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.