A114655 → θ+1 → (θ+1)²

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Triangle begins:
   1:    2;
   2:    4,     16;
   3:    8,     96,      54;
   4:   16,    384,     648,      128;
   5:   32,   1280,    4320,     2560,      250;
   6:   64,   3840,   21600,    25600,     7500,      432;
   7:  128,  10752,   90720,   179200,   105000,    18144,      686;
   8:  256,  28672,  338688,  1003520,   980000,   338688,    38416,    1024;
   9:  512,  73728, 1161216,  4816896,  7056000,  4064256,   921984,   73728,   1458;
  10: 1024, 184320, 3732480, 20643840, 42336000, 36578304, 13829760, 2211840, 131220, 2000;

The row polynomials satisfy (4*n^2-4*n-(8*n^4-20*n^3+14*n^2-2*n)*x+(n^6-3*n^5+3*n^4-n^3)*x^2)*P(n,x) = (8*n^2-8*n-(16*n^4-40*n^3-20*n^2+76*n-32)*x+(2*n^6-6*n^5-36*n^4+78*n^3-12*n^2-54*n+28)*x^2+(3*n^6-3*n^5-6*n^4+9*n^3-5*n+2)*x^3)*P(n-1,x) + ((16*n-8)*x-(32*n^3-16*n^2-16*n+12)*x^2+(4*n^5+2*n^4+8*n^3-6*n^2-8*n+6)*x^3-(2*n^5+n^4-4*n^3+n^2+2*n-1)*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[((8*n^2 - 8*n - (16*n^4 - 40*n^3 - 20*n^2 + 76*n - 32)*x + (2*n^6 - 6*n^5 - 36*n^4 + 78*n^3 - 12*n^2 - 54*n + 28)*x^2 + (3*n^6 - 3*n^5 - 6*n^4 + 9*n^3 - 5*n + 2)*x^3)*P[n - 1, x] + ((16*n - 8)*x - (32*n^3 - 16*n^2 - 16*n + 12)*x^2 + (4*n^5 + 2*n^4 + 8*n^3 - 6*n^2 - 8*n + 6)*x^3 - (2*n^5 + n^4 - 4*n^3 + n^2 + 2*n - 1)*x^4)*D[P[n - 1, x], x])/(4*n^2 - 4*n - (8*n^4 - 20*n^3 + 14*n^2 - 2*n)*x + (n^6 - 3*n^5 + 3*n^4 - n^3)*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] = ((8*n^2 - 8*n - (16*n^4 - 40*n^3 - 20*n^2 + 76*n - 32)*x + (2*n^6 - 6*n^5 - 36*n^4 + 78*n^3 - 12*n^2 - 54*n + 28)*x^2 + (3*n^6 - 3*n^5 - 6*n^4 + 9*n^3 - 5*n + 2)*x^3)*v[i - 1] + ((16*n - 8)*x - (32*n^3 - 16*n^2 - 16*n + 12)*x^2 + (4*n^5 + 2*n^4 + 8*n^3 - 6*n^2 - 8*n + 6)*x^3 - (2*n^5 + n^4 - 4*n^3 + n^2 + 2*n - 1)*x^4)*deriv(v[i - 1], x))/(4*n^2 - 4*n - (8*n^4 - 20*n^3 + 14*n^2 - 2*n)*x + (n^6 - 3*n^5 + 3*n^4 - n^3)*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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
  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 theta_plus_one of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents…?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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)
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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 → θ+1 → (θ+1)²

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.