A136572 → (θ+1)² → basis A008277

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Submission text ?

Triangle begins:
  0:        1;
  1:        4,           4;
  2:       18,          54,           18;
  3:       96,         672,          576,            96;
  4:      600,        9000,        15000,          6000,           600;
  5:     4320,      133920,       388800,        280800,         64800,         4320;
  6:    35280,     2222640,     10619280,      12348000,       4939200,       740880,        35280;
  7:   322560,    40965120,    311592960,     548674560,     338688000,     85800960,      9031680,      322560;
  8:  3265920,   832809600,   9879408000,   25376198400,   22701409920,   8641624320,   1508855040,   117573120,    3265920;
  9: 36288000, 18543168000, 338567040000, 1237602240000, 1543147200000, 828346176000, 213373440000, 27216000000, 1632960000, 36288000;

The row polynomials satisfy n*P(n,x) = (n^2+2*n+1+(n^2+2*n+1)*x)*P(n-1,x) + (n^2+2*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^2 + 2*n + 1 + (n^2 + 2*n + 1)*x)*P[n - 1, x] + (n^2 + 2*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^2 + 2*n + 1 + (n^2 + 2*n + 1)*x)*v[i - 1] + (n^2 + 2*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

Proof chains ? JSON

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

  • from theta_plus_one_squared of Triangle read by rows: row n consists of n zeros followed by n!.?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
        • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

          • from Triangle read by rows: row n consists of n zeros followed by n!.?
            • 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 polytool ? step › direct check
            • Parent-row provers of polytool polytool ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: row n consists of n zeros followed by n!.?
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Parent-row provers of polytool (above)
                • ↑ Parent-row provers of polytool (above)
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from Triangle read by rows: row n consists of n zeros followed by n!.?
                    • ↑ Iterated real-rootedness preservers (above)
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Parent-row provers of polytool (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

          • from Triangle read by rows: row n consists of n zeros followed by n!.?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
            • ↑ Laguerre’s theorem on multiplier sequences (above)
            • ↑ Laguerre’s theorem on multiplier sequences (above)
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Triangle read by rows: row n consists of n zeros followed by n!.?
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Triangle read by rows: row n consists of n zeros followed by n!.?
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from theta_plus_one_squared of Triangle read by rows: row n consists of n zeros followed by n!.?
        • ↑ Laguerre’s theorem on multiplier sequences (above)
        • ↑ Laguerre’s theorem on multiplier sequences (above)
        • ↑ Laguerre’s theorem on multiplier sequences (above)

Related ?

Same fingerprint ?

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

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. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.