A124860 → θ+1

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Triangle begins:
  0:   1;
  1:   1,    2;
  2:   3,   12,     9;
  3:   5,   30,    45,     20;
  4:  11,   88,   198,    176,     55;
  5:  21,  210,   630,    840,    525,    126;
  6:  43,  516,  1935,   3440,   3225,   1548,    301;
  7:  85, 1190,  5355,  11900,  14875,  10710,   4165,   680;
  8: 171, 2736, 14364,  38304,  59850,  57456,  33516, 10944,  1539;
  9: 341, 6138, 36828, 114576, 214830, 257796, 200508, 98208, 27621, 3410;

The row polynomials satisfy (1+(2*n-1)*x+(n^2-n)*x^2)*P(n,x) = (2+(4*n+6)*x+(2*n^2+10*n+6)*x^2+(4*n^2+8*n+2)*x^3+(2*n^2+2*n)*x^4)*P(n-2,x) + (1+(2*n+1)*x+(n^2+2*n-1)*x^2+(n^2-1)*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (4*n + 6)*x + (2*n^2 + 10*n + 6)*x^2 + (4*n^2 + 8*n + 2)*x^3 + (2*n^2 + 2*n)*x^4)*P[n - 2, x] + (1 + (2*n + 1)*x + (n^2 + 2*n - 1)*x^2 + (n^2 - 1)*x^3)*P[n - 1, x])/(1 + (2*n - 1)*x + (n^2 - n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((2 + (4*n + 6)*x + (2*n^2 + 10*n + 6)*x^2 + (4*n^2 + 8*n + 2)*x^3 + (2*n^2 + 2*n)*x^4)*v[i - 2] + (1 + (2*n + 1)*x + (n^2 + 2*n - 1)*x^2 + (n^2 - 1)*x^3)*v[i - 1])/(1 + (2*n - 1)*x + (n^2 - n)*x^2)); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from 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 A Jacobsthal-Pascal triangle.?
    • 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 A Jacobsthal-Pascal triangle.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

• A344565 → θ+1
• A038208 → θ+1
• A038221 → D+1
• A085880 → basis A013609 → θ+1
• A178820 → D+1
• A124860 → D+1
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
• A135065 → D+1
• A003506 → D → θ+1
• A178820 → basis A013609 → D+1
• A093375 → D → D
• A038338 → θ+1
• A135838 → θ+1
• A131431 → x ↦ x+1 → θ+1
• A156992 → D+1
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
• basis-map coefficients of A113214, shifted → basis A013609 → θ+1
• A028326 → D+1
• A356546 → basis A013609 → D+1
• basis-map coefficients of A113214, shifted → basis A013609 → D+1
• A038208 → D+1
• A085880 → D → θ+1
• A178822 → basis A013609 → θ+1
• A196347 → basis A013609 → θ+1
• basis-map coefficients of A026729 → basis A007318 → θ+1
• A028326 → D → θ+1
• A135065 → θ+1
• A178821 → θ+1
• basis-map coefficients of A113214, shifted → D+1
• A120910 → basis A013610 → D+1
• A178822 → D+1
• A376827 → D+1
• A038221 → basis A013609 → D+1
• A028326 → D → D+1
• A085880 → θ+1
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• A085880 → D+1
• basis-map coefficients of A113214, shifted → D → θ+1
• A038330 → basis A013611 → D+1
• A135838 → D+1
• A028326 → basis A013609 → θ+1
• A038221 → θ+1
• A120910 → basis A013610 → θ+1
• A344565 → D+1
• A376827 → θ+1
• A038221 → basis A013609 → θ+1
• A038325 → D+1
• A094305 → θ+1
• A196347 → θ+1
• A085880 → basis A013609 → D+1
• A038338 → basis A013609 → D+1
• A356546 → θ+1
• basis-map coefficients of A113214, shifted → D → D+1
• A156992 → θ+1
• basis-map coefficients of A026729 → basis A007318 → D+1
• A134346 → θ+1
• A038325 → basis A013609 → θ+1
• A216973 → θ+1
• A094305 → D+1
• A038325 → θ+1
• A093375 → D
• A356546 → basis A013609 → θ+1
• basis-map coefficients of A026729, shifted → basis A007318 → D+1
• A038338 → D+1
• A178822 → θ+1
• basis-map coefficients of A127647 → D+1
• A131431 → x ↦ x+1 → D+1
• A178821 → D+1
• A178820 → θ+1
• A178821 → basis A013609 → θ+1
• A038208 → D → D+1
• A085880 → D → D+1
• A131431 → D+1 → basis A007318
• A028326 → θ+1
• A038330 → basis A013611 → θ+1
• A038208 → D → θ+1
• basis-map coefficients of A112227 → x ↦ x+1 → D+1
• basis-map coefficients of A026729, shifted → basis A007318 → θ+1
• basis-map coefficients of A127647 → θ+1
• A156992 → basis A013609 → D+1
• A027465 → basis A013611 → D+1
• A356546 → D+1
• A085881 → D+1
• A085881 → basis A013609 → D+1
• A196347 → D+1
• A038325 → basis A013609 → D+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• A085881 → basis A013609 → θ+1
• A027465 → basis A013611 → θ+1
• basis-map coefficients of A112227 → x ↦ x+1 → θ+1
• A178821 → basis A013609 → D+1
• A028326 → basis A013609 → D+1
• A178822 → basis A013609 → D+1
• A038338 → basis A013609 → θ+1
• A134346 → D+1
• A178820 → basis A013609 → θ+1
• A196347 → basis A013609 → D+1
• basis-map coefficients of A113214, shifted → θ+1
• A085881 → θ+1
• A156992 → basis A013609 → θ+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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.