A124860 → D+1

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Submission text ?

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

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 2*n + (4*n^2 + 8*n + 2)*x + (2*n^2 + 10*n + 6)*x^2 + (4*n + 6)*x^3 + 2*x^4)*v[i - 2] + (n^2 - 1 + (n^2 + 2*n - 1)*x + (2*n + 1)*x^2 + x^3)*v[i - 1])/(n^2 - n + (2*n - 1)*x + 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 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 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 ?

• A196347 → D+1
• A085880 → D → D+1
• A131431 → x ↦ x+1 → θ+1
• A120910 → basis A013610 → D+1
• A003506 → D → θ+1
• A028326 → D+1
• A178820 → basis A013609 → θ+1
• A356546 → basis A013609 → θ+1
• A038330 → basis A013611 → D+1
• A038221 → D+1
• A124860 → θ+1
• A038208 → θ+1
• basis-map coefficients of A026729 → basis A007318 → θ+1
• basis-map coefficients of A113214, shifted → D+1
• A376827 → D+1
• A085880 → basis A013609 → D+1
• A038338 → θ+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• A085880 → D+1
• A038208 → D+1
• A038208 → D → θ+1
• A196347 → basis A013609 → D+1
• A027465 → basis A013611 → θ+1
• A028326 → basis A013609 → D+1
• A038325 → θ+1
• A038325 → D+1
• A134346 → D+1
• basis-map coefficients of A112227 → x ↦ x+1 → D+1
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
• A131431 → x ↦ x+1 → D+1
• A156992 → basis A013609 → D+1
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
• A038338 → D+1
• A344565 → θ+1
• A038330 → basis A013611 → θ+1
• A131431 → D+1 → basis A007318
• A178821 → θ+1
• A028326 → θ+1
• A028326 → basis A013609 → θ+1
• A038338 → basis A013609 → D+1
• A178821 → basis A013609 → θ+1
• A028326 → D → D+1
• A085880 → basis A013609 → θ+1
• A038208 → D → D+1
• A085881 → D+1
• A027465 → basis A013611 → D+1
• A120910 → basis A013610 → θ+1
• A356546 → θ+1
• A085880 → D → θ+1
• basis-map coefficients of A113214, shifted → basis A013609 → D+1
• A085881 → θ+1
• A178822 → D+1
• A196347 → θ+1
• A178820 → basis A013609 → D+1
• A094305 → D+1
• A178822 → θ+1
• A196347 → basis A013609 → θ+1
• A038338 → basis A013609 → θ+1
• A134346 → θ+1
• A156992 → D+1
• A356546 → D+1
• A085881 → basis A013609 → θ+1
• A028326 → D → θ+1
• A094305 → θ+1
• A093375 → D → D
• A038221 → θ+1
• A135838 → D+1
• basis-map coefficients of A113214, shifted → D → θ+1
• A376827 → θ+1
• A135065 → D+1
• A178822 → basis A013609 → D+1
• A178820 → D+1
• A156992 → θ+1
• A038221 → basis A013609 → θ+1
• basis-map coefficients of A112227 → x ↦ x+1 → θ+1
• A135065 → θ+1
• A085880 → θ+1
• A085881 → basis A013609 → D+1
• basis-map coefficients of A113214, shifted → basis A013609 → θ+1
• basis-map coefficients of A113214, shifted → D → D+1
• basis-map coefficients of A127647 → θ+1
• A038221 → basis A013609 → D+1
• A356546 → basis A013609 → D+1
• basis-map coefficients of A026729, shifted → basis A007318 → θ+1
• basis-map coefficients of A127647 → D+1
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• A038325 → basis A013609 → D+1
• basis-map coefficients of A113214, shifted → θ+1
• A216973 → θ+1
• A344565 → D+1
• A135838 → θ+1
• A178822 → basis A013609 → θ+1
• A178821 → basis A013609 → D+1
• A038325 → basis A013609 → θ+1
• basis-map coefficients of A026729, shifted → basis A007318 → D+1
• A156992 → basis A013609 → θ+1
• A093375 → D
• A178821 → D+1
• basis-map coefficients of A026729 → basis A007318 → D+1
• A178820 → 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.