A134082 → basis A007318 → x ↦ x+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:    1;
  1:    4,     1;
  2:   12,     8,     1;
  3:   32,    36,    12,     1;
  4:   80,   128,    72,    16,     1;
  5:  192,   400,   320,   120,    20,     1;
  6:  448,  1152,  1200,   640,   180,    24,    1;
  7: 1024,  3136,  4032,  2800,  1120,   252,   28,   1;
  8: 2304,  8192, 12544, 10752,  5600,  1792,  336,  32,  1;
  9: 5120, 20736, 36864, 37632, 24192, 10080, 2688, 432, 36, 1;

The row polynomials satisfy P(n,x) = -(2*n-6-x)*P(n-1,x) + (4+2*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = -(2*n - 6 - x)*v[i - 1] + (4 + 2*x)*deriv(v[i - 1], x)); 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)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from shift_x_plus_one of Triangle read by rows, (n-1) zeros followed by (2n, 1).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows, (n-1) zeros followed by (2n, 1).?
        • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

          • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

              • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                • 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
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Iterated real-rootedness preservers (above)
            • Parent-row provers of polytool polytool ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Veronese sections of Pólya frequency sequences (above)
                • ↑ Parent-row provers of polytool (above)
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows, (n-1) zeros followed by (2n, 1).?
        • ↑ Laguerre’s theorem on differential operators (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows, (n-1) zeros followed by (2n, 1).?
        • ↑ Laguerre’s theorem on differential operators (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Candidate equalities ?

• A116666 → reverse → x ↦ x+1 15 rows; no all-index equality certificate was found

Same fingerprint ?

• A013609 → D → θ+1
• A116666 → x ↦ x/(x+1)
• A356546 → basis A013611 → θ+1
• A120909 → θ+1
• A196347 → basis A013611 → θ+1
• A178821 → basis A013611 → θ+1
• A038221 → basis A013611 → θ+1
• basis-map coefficients of A112227 → basis A013609 → θ+1
• A130123 → basis A013609 → θ+1
• A178822 → basis A013611 → θ+1
• A178820 → basis A013611 → θ+1
• A038208 → basis A013611 → θ+1
• A013609 → θ+1
• basis-map coefficients of A026729, shifted → basis A013609 → θ+1
• A013609 → basis A013610 → θ+1
• A120909 → basis A013610 → θ+1
• A156992 → basis A013611 → θ+1
• basis-map coefficients of A026729 → basis A013609 → θ+1
• A028326 → basis A013611 → θ+1
• A085881 → basis A013611 → θ+1
• A085880 → basis A013611 → θ+1
• A136572 → basis A013609 → θ+1
• basis-map coefficients of A030528, shifted → basis A013609 → θ+1

References

1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
3. [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
4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
6. [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
7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.