A132440 → θ+1 → basis A013610

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Submission text ?

Triangle begins:
  0:  0;
  1:  1;
  2:  4,   12;
  3:  9,   54,    81;
  4: 16,  144,   432,    432;
  5: 25,  300,  1350,   2700,   2025;
  6: 36,  540,  3240,   9720,  14580,    8748;
  7: 49,  882,  6615,  26460,  59535,   71442,   35721;
  8: 64, 1344, 12096,  60480, 181440,  326592,  326592,  139968;
  9: 81, 1944, 20412, 122472, 459270, 1102248, 1653372, 1417176, 531441;

The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n+(6*n^2-6*n)*x+(9*n^2-9*n)*x^2)*P(n-2,x) + (2*n^2-4*n+(6*n^2-12*n)*x)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 4 + 12*x.

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 4 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n + (6*n^2 - 6*n)*x + (9*n^2 - 9*n)*x^2)*P[n - 2, x] + (2*n^2 - 4*n + (6*n^2 - 12*n)*x)*P[n - 1, x])/(n^2 - 3*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 4 + 12*x; for(n = 3, N, my(i = n + 1); v[i] = (-(n^2 - n + (6*n^2 - 6*n)*x + (9*n^2 - 9*n)*x^2)*v[i - 2] + (2*n^2 - 4*n + (6*n^2 - 12*n)*x)*v[i - 1])/(n^2 - 3*n + 2)); 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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from theta_plus_one of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix,…?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical ope…?
        • 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)
        • 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 (above)
            • ↑ Veronese sections of Pólya frequency sequences (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 Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical ope…?
        • ↑ Laguerre’s theorem on differential operators (above)
        • ↑ Laguerre’s theorem on differential operators (above)

Related ?

Same fingerprint ?

• A013609 → basis A013610 → x ↦ x/(x+1)
• basis-map coefficients of A026729, shifted → θ+1 → basis A013610
• A196347 → x ↦ x+1 → basis A007318
• A038279 → basis A013610
• A136572 → basis A013610 → basis A013611
• A156992 → basis A013611 → x ↦ x/(x+1)
• A028326 → basis A007318 → basis A007318
• basis-map coefficients of A026729, shifted → basis A013610
• A120910 → basis A007318
• A129186 → basis A013610
• A130123 → basis A013610 → D
• basis-map coefficients of A026729 → D → basis A013610
• A178822 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A112227 → θ+1 → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013611
• A130123 → basis A013610
• A127648 → D → basis A013610
• image under θ+1 → (θ+1)² → basis A013610
• A136572 → θ+1 → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610 → reverse
• A130123 → basis A013610 → basis A013611
• basis-map coefficients of A026729 → basis A013610 → basis A013611
• A136572 → (θ+1)² → basis A013610
• A038338 → x ↦ x+1 → basis A007318
• A013610 → basis A013611
• A038329 → x ↦ x+1 → basis A013609
• basis-map coefficients of A026729 → basis A013610
• basis-map coefficients of A112227 → basis A013610
• A038243 → basis A013609 → D
• basis-map coefficients of A021012 → θ+1 → basis A013610
• A120909 → basis A013610 → x ↦ x/(x+1)
• A356546 → basis A013611 → x ↦ x/(x+1)
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
• A085881 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A026729 → basis A013610 → reverse
• A085880 → x ↦ x+1 → x ↦ x+1
• A085880 → basis A013611 → x ↦ x/(x+1)
• A120910 → basis A007318 → reverse
• A134309 → basis A013610
• A013610 → D → basis A013611
• A013610 → basis A013611 → D
• A027465: Cube of lower triangular normalized binomial matrix.
• A131431 → D → basis A013610
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013610
• A178821 → basis A013611 → x ↦ x/(x+1)
• A196347 → basis A013611 → x ↦ x/(x+1)
• A356546 → x ↦ x+1 → x ↦ x+1
• basis-map coefficients of A112227 → (θ+1)² → basis A013610
• A136572 → basis A013610 → D
• basis-map coefficients of A112227 → basis A013610 → basis A013611
• A038279 → basis A013610 → reverse
• basis-map coefficients of A026729 → basis A013610 → D
• A120910 → D → x ↦ x+1
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013610
• basis-map coefficients of A026729, shifted → D → basis A013610
• A136521 → basis A013610
• A178822 → x ↦ x+1 → x ↦ x+1
• A120909 → x ↦ x/(x+1)
• A094305 → x ↦ x+1 → x ↦ x+1
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
• basis-map coefficients of A030528, shifted → D → basis A013610
• image under θ+1 → θ+1 → D → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610 → D
• A130123 → (θ+1)² → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → reverse
• A136572 → basis A013610
• A013610 → D → D
• A027465 → D → D
• A131436 → basis A013610
• A130123 → D → basis A013610
• A131431 → θ+1 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610
• basis-map coefficients of A112227 → basis A013610 → D
• A028326 → basis A013611 → x ↦ x/(x+1)
• A131431 → basis A013610
• A038325 → basis A007318 → basis A007318
• A130123 → θ+1 → basis A013610
• A038330 → basis A013609 → x ↦ x+1
• A038279 → basis A013610 → D
• A178820 → basis A013611 → x ↦ x/(x+1)
• A013610 → basis A013611 → reverse
• image under θ+1 → basis A013610
• basis-map coefficients of A026729 → (θ+1)² → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → D
• A135065 → x ↦ x+1 → basis A007318
• A178820 → basis A007318 → basis A007318
• A038330 → reverse
• basis-map coefficients of A112227 → basis A013610 → reverse
• A085881 → basis A007318 → basis A007318
• basis-map coefficients of A021012 → basis A013610
• A147716 → basis A013610 → basis A013609
• A038221 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A030528, shifted → θ+1 → basis A013610
• A127648 → basis A013610
• A013610 → D
• A027465 → D
• A038243 → basis A013609
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013611
• A156992 → basis A007318 → x ↦ x+1
• A178821 → basis A007318 → x ↦ x+1
• A136572 → D → basis A013610
• A038330 → D

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