A028326 → basis A013610 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:       2;
  1:      10,        6;
  2:      50,       60,       18;
  3:     250,      450,      270,       54;
  4:    1250,     3000,     2700,     1080,      162;
  5:    6250,    18750,    22500,    13500,     4050,      486;
  6:   31250,   112500,   168750,   135000,    60750,    14580,     1458;
  7:  156250,   656250,  1181250,  1181250,   708750,   255150,    51030,    4374;
  8:  781250,  3750000,  7875000,  9450000,  7087500,  3402000,  1020600,  174960,  13122;
  9: 3906250, 21093750, 50625000, 70875000, 63787500, 38272500, 15309000, 3936600, 590490, 39366;

The row polynomials satisfy P(n,x) = (5+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 2.

P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(5 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = (5 + 3*x)*v[i - 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013610 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
    • Geometric basis map (elementary) 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)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from shift_x_plus_one of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
        • 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)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
        • Geometric basis map (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)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013610 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A013612 → basis A013609 → reverse
• A038338 → basis A013610 → x ↦ x+1
• A038221 → basis A013610 → basis A007318
• A013620 → basis A007318 → D
• A038329 → basis A013610 → D
• A120910 → basis A013609 → x ↦ x/(x+1)
• A178821 → basis A013610 → x ↦ x+1
• A038284 → basis A007318 → basis A013609
• A038255 → basis A013610 → basis A013609
• A013622 → D
• A027467 → basis A013611 → basis A013610
• A013612 → basis A013609 → D
• A038334 → x ↦ x/(x+1)
• A013622 → basis A013609 → basis A013610
• A038329 → basis A013610
• A038271 → basis A013611
• A038243 → basis A013611 → x ↦ x/(x+1)
• A156992 → basis A013610 → x ↦ x+1
• A038271 → basis A013610 → basis A013610
• A178820 → basis A013610 → x ↦ x+1
• A013622: Triangle of coefficients in expansion of (3+5x)^n.
• A196347 → basis A013610 → basis A007318
• A038329 → basis A013610 → reverse
• A038271 → basis A013611 → D
• A038231 → D → basis A013610
• A085880 → basis A013610 → x ↦ x+1
• A038335 → x ↦ x+1 → basis A013609
• A013620 → basis A007318
• A178822 → basis A013610 → basis A007318
• A013612 → basis A013609
• A038319 → basis A013611 → 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.