A048993 → D+1 → θ+1

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Submission text ?

Triangle begins:
  0: 1;
  1: 1,    2;
  2: 1,    6,     3;
  3: 1,   14,    18,      4;
  4: 1,   30,    75,     40,      5;
  5: 1,   62,   270,    260,     75,      6;
  6: 1,  126,   903,   1400,    700,    126,     7;
  7: 1,  254,  2898,   6804,   5250,   1596,   196,    8;
  8: 1,  510,  9075,  31080,  34755,  15876,  3234,  288,   9;
  9: 1, 1022, 27990, 136420, 212625, 136962, 41160, 6000, 405, 10;

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

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - x^2*D[P[n - 2, x], x] + (1 + 2*x)*P[n - 1, x] + 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; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] - x^2*deriv(v[i - 2], x) + (1 + 2*x)*v[i - 1] + 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 (−∞, 0]
  by Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A008277 of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.?
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

          • from basis-map coefficients of Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.?
            • 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 [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 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) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.?
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.?
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Ray basis map theorem (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? 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) (above)

Related ?

Same fingerprint ?

• A008277 → θ+1
• A321331: Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).
• basis-map coefficients of A021012 → basis A008277 → θ+1
• A130123 → basis A008277 → θ+1
• basis-map coefficients of A112227 → basis A008277 → θ+1
• A008277 → θ+1 → reverse

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.