image under θ+1 → basis A008277

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Triangle begins:
  0:  1;
  1:  2,    2;
  2:  3,    9,     3;
  3:  4,   28,    24,      4;
  4:  5,   75,   125,     50,      5;
  5:  6,  186,   540,    390,     90,      6;
  6:  7,  441,  2107,   2450,    980,    147,     7;
  7:  8, 1016,  7728,  13608,   8400,   2128,   224,    8;
  8:  9, 2295, 27225,  69930,  62559,  23814,  4158,  324,   9;
  9: 10, 5110, 93300, 341050, 425250, 228270, 58800, 7500, 450, 10;

The row polynomials satisfy n*P(n,x) = (n+1+(n+1)*x)*P(n-1,x) + (n+1)*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[Together[((n + 1 + (n + 1)*x)*P[n - 1, x] + (n + 1)*x*D[P[n - 1, x], x])/(n)]];
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] = ((n + 1 + (n + 1)*x)*v[i - 1] + (n + 1)*x*deriv(v[i - 1], x))/(n)); 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 Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem
• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem

Proof chains ? JSON

• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

  • from derivative of basis-map coefficients of A scaled Hermite triangle.?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from basis-map coefficients of A scaled Hermite triangle.?
        • 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 A scaled Hermite triangle.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

  • from theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
    • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Laguerre’s theorem on multiplier sequences (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A026729 → (θ+1)² → basis A008277
• A131431 → θ+1 → basis A008277
• basis-map coefficients of A021012 → θ+1 → basis A008277
• A131431 → basis A008277
• basis-map coefficients of A030528, shifted → D → basis A008277
• A136572 → D → basis A008277
• A130123 → basis A008277
• basis-map coefficients of A021012 → basis A008277 → reverse
• A129186 → basis A008277
• basis-map coefficients of A026729, shifted → D → basis A008277
• A134309 → basis A008277
• basis-map coefficients of A030528, shifted → basis A008277
• basis-map coefficients of A030528, shifted → θ+1 → basis A008277
• A130123 → basis A008277 → reverse
• image under θ+1 → θ+1 → D → basis A008277
• basis-map coefficients of A026729 → D → basis A008277
• A130123 → D → basis A008277
• basis-map coefficients of A021012 → basis A008277
• basis-map coefficients of A112227 → basis A008277
• A130123 → (θ+1)² → basis A008277
• A048993 → reverse
• A130123 → θ+1 → basis A008277
• A008278: Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
• A132440 → basis A008277
• A008277: Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
• A048993: Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.
• A136572 → (θ+1)² → basis A008277
• image under θ+1 → (θ+1)² → basis A008277
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A008277
• A136572 → basis A008277
• A127648 → D → basis A008277
• A131431 → D → basis A008277
• basis-map coefficients of A112227 → basis A008277 → reverse
• basis-map coefficients of A112227 → θ+1 → basis A008277
• A132440 → θ+1 → basis A008277
• A048993 → D+1 → reverse
• A106800: Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
• A136521 → basis A008277
• A048993 → D+1
• A136572 → θ+1 → basis A008277
• basis-map coefficients of A112227 → (θ+1)² → basis A008277
• A131436 → basis A008277
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A008277
• basis-map coefficients of A026729, shifted → θ+1 → basis A008277
• basis-map coefficients of A026729, shifted → basis A008277

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.