A028326 → D → basis A013610

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Triangle begins:
  0:    0;
  1:    2;
  2:    8,    12;
  3:   24,    72,     54;
  4:   64,   288,    432,    216;
  5:  160,   960,   2160,   2160,     810;
  6:  384,  2880,   8640,  12960,    9720,    2916;
  7:  896,  8064,  30240,  60480,   68040,   40824,   10206;
  8: 2048, 21504,  96768, 241920,  362880,  326592,  163296,  34992;
  9: 4608, 55296, 290304, 870912, 1632960, 1959552, 1469664, 629856, 118098;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 3*n*x)*v[i - 1])/(n - 1)); 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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from derivative of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
        • 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 Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • Parent-row provers of polytool polytool ? step › direct check
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.?
        • 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
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle read by rows: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.?
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

• A038325 → basis A013609 → basis A013610
• A134346 → basis A013610
• basis-map coefficients of A026729 → basis A013609 → x ↦ x+1
• A131431 → x ↦ x+1 → basis A013610
• A156992 → basis A007318 → basis A013609
• A038334 → reverse
• A013609 → basis A013610 → x ↦ x+1
• A178821 → basis A013610
• basis-map coefficients of A113214, shifted → basis A013610
• A135065 → x ↦ x+1 → x ↦ x/(x+1)
• A178821 → basis A013609 → basis A013610
• A196347 → basis A013611 → x ↦ x+1
• A216973 → basis A013610
• A120909 → basis A013610 → x ↦ x+1
• A038338 → basis A013610 → reverse
• A178822 → basis A013611 → basis A007318
• A038330 → basis A013611 → basis A013610
• A013610 → basis A013611 → basis A013609
• A156992 → basis A013610
• A134058 → basis A013610
• A178820 → basis A013609 → basis A013610
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A038330 → basis A013609 → x ↦ x/(x+1)
• A038221 → basis A013609 → basis A013610
• A085880 → basis A013610 → D
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A124860 → basis A013610
• basis-map coefficients of A026729, shifted → basis A007318 → basis A013610
• A038338 → basis A013609 → basis A013610
• A178820 → basis A013611 → basis A007318
• A178822 → basis A013609 → basis A013610
• A038208 → D → basis A013610
• A178820 → basis A013610
• A038338 → basis A013610
• A038334 → D
• A094305 → x ↦ x+1 → x ↦ x/(x+1)
• A356546 → basis A013611 → basis A007318
• A085880 → D → basis A013610
• A085881 → basis A013611 → basis A007318
• A085881 → basis A007318 → basis A013609
• A178822 → basis A013610 → D
• A356546 → x ↦ x+1 → basis A013609
• A028326 → basis A007318 → basis A013609
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• basis-map coefficients of A030528, shifted → basis A013609 → x ↦ x+1
• A120910 → x ↦ x/(x+1)
• A003506 → x ↦ x+1 → basis A013609
• A135065 → basis A013610
• A085881 → basis A013609 → basis A013610
• A178820 → basis A007318 → basis A013609
• A134059 → basis A013610
• A178822 → basis A013610
• A085880 → basis A013609 → basis A013610
• A120910 → basis A013610 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013609
• A038291 → basis A013609 → basis A013611
• A094305 → x ↦ x+1 → basis A013609
• A038243 → basis A013611
• A038220 → D
• A135065 → x ↦ x+1 → basis A013609
• A038243 → basis A013611 → D
• A136572 → basis A013609 → x ↦ x+1
• A156992 → basis A013610 → D
• A120910 → D → basis A013609
• A178822 → x ↦ x+1 → basis A013609
• A178820 → basis A013610 → D
• A196347 → basis A013610 → D
• A178821 → basis A013610 → reverse
• A196347 → basis A013610
• basis-map coefficients of A113214, shifted → D → basis A013610
• A038208 → basis A013610 → D
• A135089 → basis A013610
• A120910 → basis A013609 → D
• basis-map coefficients of A026729 → basis A013610 → basis A013609
• A356546 → basis A013610
• basis-map coefficients of A112227 → x ↦ x+1 → basis A013610
• A344565 → basis A013610
• basis-map coefficients of A127647 → basis A013610
• A028326 → basis A013610 → D
• A038221 → basis A013610 → D
• A013620 → D
• A038221 → basis A013611 → x ↦ x+1
• A013610 → D → basis A013609
• A038325 → basis A013610
• A085880 → basis A013610
• A038208 → basis A013610
• A156992 → basis A013609 → basis A013610
• A085880 → basis A013610 → reverse
• A135838 → basis A013610
• A356546 → basis A013609 → basis A013610
• basis-map coefficients of A112227 → basis A013609 → x ↦ x+1
• A028326 → basis A013611 → x ↦ x+1
• A376827 → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013609
• A178821 → basis A013610 → D
• A136572 → basis A013610 → basis A013609
• A038221 → basis A013610
• A120910 → basis A013609 → reverse
• basis-map coefficients of A026729, shifted → basis A013609 → basis A007318
• A120910 → basis A013609
• basis-map coefficients of A026729 → basis A007318 → basis A013610
• A027466 → basis A013611 → x ↦ x/(x+1)
• A178821 → basis A013611 → basis A007318
• A038328 → basis A013609 → basis A013610
• A003506 → D → basis A013610
• A156992 → basis A013611 → x ↦ x+1
• A038330 → basis A013609 → basis A013609
• A038338 → basis A013610 → D
• A028326 → basis A013609 → basis A013610
• A028326 → basis A013610
• basis-map coefficients of A112227 → basis A013610 → basis A013609
• A038325 → basis A007318 → basis A013609
• A027465 → basis A013611 → basis A013610
• A094305 → basis A013610
• A085880 → basis A013611 → x ↦ x+1
• A258758 → basis A013610
• A085881 → basis A013610
• A132200 → basis A013610

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.