A136572 → basis A013609 → D

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Submission text ?

Triangle begins:
  0:       0;
  1:       2;
  2:       8,        16;
  3:      36,       144,       144;
  4:     192,      1152,      2304,       1536;
  5:    1200,      9600,     28800,      38400,      19200;
  6:    8640,     86400,    345600,     691200,     691200,      276480;
  7:   70560,    846720,   4233600,   11289600,   16934400,    13547520,     4515840;
  8:  645120,   9031680,  54190080,  180633600,  361267200,   433520640,   289013760,   82575360;
  9: 6531840, 104509440, 731566080, 2926264320, 7315660800, 11705057280, 11705057280, 6688604160, 1672151040;

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

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

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

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.?
    • Geometric basis map (elementary) 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)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A007318 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
            • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • Geometric basis map (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)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Geometric basis map (elementary) (above)
                • Geometric basis map (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)
                    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                • 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) (above)
                    • ↑ Affine, reversal and Möbius substitutions (elementary) (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)
                    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Geometric basis map (elementary) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Geometric basis map (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Geometric basis map (elementary) (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 basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.?
            • ↑ Geometric basis map (elementary) (above)
            • Geometric basis map (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)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Geometric basis map (elementary) (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                • ↑ Geometric basis map (elementary) (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 Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Geometric basis map (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Geometric basis map (elementary) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A356546 → x ↦ x+1
• A178821 → basis A013611 → basis A013610
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
• basis-map coefficients of A113214, shifted → D → basis A013611
• A003506 → D → basis A013611
• A038221 → basis A013611
• A085880 → basis A013611 → basis A013610
• A120910 → D
• A085880 → basis A013609 → basis A013611
• A124860 → basis A013611
• A178822 → x ↦ x+1 → D
• A134346 → x ↦ x/(x+1)
• A131431 → x ↦ x+1 → x ↦ x/(x+1)
• A085880 → D → basis A013611
• basis-map coefficients of A113214, shifted → basis A013609 → basis A013611
• A038338 → basis A013609 → basis A007318
• basis-map coefficients of A113214, shifted → D → basis A007318
• A376827 → x ↦ x/(x+1)
• A135838 → x ↦ x+1
• A376827 → basis A013611
• A134059 → basis A013611
• A013609 → basis A013610
• A085880 → basis A013611
• A258758 → basis A007318
• A356546 → basis A013611 → D
• A038208 → basis A007318
• A136572 → basis A013609
• A156992 → basis A013609 → basis A007318
• A131436 → basis A013609
• A258758 → basis A013611
• A178822 → basis A013609 → x ↦ x+1
• A159854 → x ↦ x/(x+1)
• A131431 → D → basis A013609
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013609
• basis-map coefficients of A113214, shifted → basis A013609 → x ↦ x+1
• A131431 → basis A013609
• A376827 → basis A007318
• A038208 → basis A013611 → D
• Veronese parent of A098172 → x ↦ x/(x+1)
• basis-map coefficients of A026729 → basis A007318 → basis A013611
• A130123 → basis A013609 → D
• A038221 → basis A013611 → D
• A216973 → basis A013611
• Veronese parent of A098172 → x ↦ x/(x+1)
• A120910 → basis A013610 → basis A013611
• basis-map coefficients of A112227 → basis A013609 → D
• A038338 → x ↦ x+1 → D
• basis-map coefficients of A026729, shifted → D → basis A013609
• A120909 → basis A013610 → reverse
• basis-map coefficients of A026729, shifted → θ+1 → basis A013609
• A013609 → D
• A038208 → D → basis A013611
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013609
• A135065 → basis A013611
• A156992 → basis A013611 → basis A013610
• A356546 → x ↦ x+1 → D
• A132440 → θ+1 → basis A013609
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
• basis-map coefficients of A026729, shifted → basis A007318 → basis A013611
• A028326 → basis A007318 → reverse
• A134309 → basis A013609
• A085880 → basis A013609 → x ↦ x+1
• A120909 → basis A013610 → D
• A085880 → x ↦ x+1 → reverse
• basis-map coefficients of A113214, shifted → x ↦ x+1
• A120910 → D → D
• A085881 → basis A013611 → D
• A094305 → x ↦ x+1 → reverse
• A003506 → x ↦ x+1
• A136572 → (θ+1)² → basis A013609
• A178820 → x ↦ x/(x+1)
• A129186 → basis A013609
• A135065 → x ↦ x+1
• A136572 → basis A013609 → basis A013610
• A132440 → basis A013609
• A085881 → basis A013611 → reverse
• A178821 → basis A013611 → reverse
• A038338 → x ↦ x+1
• A356546 → basis A013611 → basis A013610
• A178822 → x ↦ x/(x+1)
• A003506 → x ↦ x/(x+1)
• basis-map coefficients of A026729 → basis A013609 → D
• A028326 → basis A013611
• A120909 → D
• A094305 → x ↦ x+1
• A178820 → basis A013611 → D
• A178821 → basis A013611 → D
• A085880 → x ↦ x+1
• basis-map coefficients of A112227 → basis A013609
• A038325 → basis A007318
• A134058 → basis A007318
• A178822 → x ↦ x+1
• A120910 → basis A007318 → basis A013609
• A196347 → basis A013609 → basis A007318
• basis-map coefficients of A030528, shifted → basis A013609
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013609 → D
• A085880 → basis A013611 → D
• A120909 → basis A013610
• A127648 → D → basis A013609
• basis-map coefficients of A026729, shifted → basis A007318 → x ↦ x+1
• A038325 → basis A007318 → reverse
• A178822 → basis A013611 → basis A013610
• basis-map coefficients of A021012 → θ+1 → basis A013609
• A196347 → basis A013611
• A130123 → D → basis A013609
• A156992 → basis A013611 → D
• basis-map coefficients of A112227 → basis A013609 → basis A013610
• A178822 → basis A013609 → basis A013611
• A178820 → basis A013611 → basis A013610
• basis-map coefficients of A127647 → x ↦ x/(x+1)
• A013609 → basis A013610 → basis A013610
• basis-map coefficients of A113214, shifted → basis A013611
• A135089 → basis A013611
• A094305 → x ↦ x/(x+1)
• A013609 → D → D
• A134058 → basis A013611
• A085881 → basis A013611
• A156992 → basis A013611
• A038330 → basis A013609
• A028326 → basis A013609 → x ↦ x+1
• A085880 → D → basis A007318
• A028326 → basis A007318 → D
• A156992 → x ↦ x/(x+1)
• basis-map coefficients of A026729 → (θ+1)² → basis A013609
• A135838 → basis A013611
• A356546 → basis A013609 → basis A013611
• A131431 → x ↦ x+1 → basis A007318
• basis-map coefficients of A026729 → D → basis A013609
• A130123 → θ+1 → basis A013609
• A085880 → x ↦ x/(x+1)
• A038208 → basis A007318 → D
• A028326 → basis A007318
• A196347 → basis A013611 → basis A013610
• A124860 → x ↦ x+1
• A178820 → basis A013609 → basis A013611
• A120909 → basis A013610 → basis A013610
• A038338 → x ↦ x+1 → reverse
• A135065 → x ↦ x+1 → reverse
• A136572 → D → basis A013609
• basis-map coefficients of A112227 → x ↦ x+1 → basis A007318
• A178822 → basis A013611
• A038330 → basis A013609 → D
• basis-map coefficients of A030528, shifted → D → basis A013609
• A028326 → basis A013611 → basis A013610
• A178821 → basis A007318 → reverse
• A132200 → basis A013611
• A131431 → x ↦ x+1 → basis A013611
• A085881 → basis A007318 → reverse
• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-…
• A196347 → x ↦ x+1
• A136572 → θ+1 → basis A013609
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• A356546 → basis A013611 → reverse
• A147716 → basis A013610 → basis A013610
• A038325 → basis A013611
• A356546 → x ↦ x+1 → reverse
• A178820 → basis A013611 → reverse
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013610
• basis-map coefficients of A030528, shifted → θ+1 → basis A013609
• A131431 → θ+1 → basis A013609
• A038221 → x ↦ x+1 → D
• A178821 → basis A007318
• A178821 → basis A007318 → D
• A132200 → basis A007318
• A027466 → basis A013611 → D
• basis-map coefficients of A026729 → basis A013609 → basis A013610
• A038221 → basis A013609 → basis A013611
• A013609 → D → basis A013610
• A178821 → basis A013611
• A124860 → x ↦ x/(x+1)
• image under θ+1 → (θ+1)² → basis A013609
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
• A085881 → basis A007318 → D
• A038221 → x ↦ x+1
• basis-map coefficients of A026729 → basis A013609
• A130123 → basis A013609
• basis-map coefficients of A127647 → basis A013611
• basis-map coefficients of A112227 → x ↦ x+1 → basis A013611
• A356546 → x ↦ x/(x+1)
• image under θ+1 → θ+1 → D → basis A013609
• A134346 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013609
• basis-map coefficients of A021012 → basis A013609
• basis-map coefficients of A026729 → basis A013609 → reverse
• A196347 → x ↦ x+1 → D
• A156992 → basis A007318
• A085881 → basis A013611 → basis A013610
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• basis-map coefficients of A026729, shifted → basis A013609 → D
• A344565 → basis A007318
• A038325 → x ↦ x/(x+1)
• A085881 → x ↦ x/(x+1)
• A038291 → basis A013609 → basis A013610
• A038338 → basis A013611
• A344565 → basis A013611
• image under θ+1 → basis A013609
• A135065 → x ↦ x+1 → D
• A156992 → basis A007318 → D
• A356546 → basis A013611
• A038338 → x ↦ x/(x+1)
• A085881 → basis A007318
• basis-map coefficients of A127647 → basis A007318
• A028326 → basis A013611 → D
• basis-map coefficients of A112227 → (θ+1)² → basis A013609
• A027465 → basis A013611 → basis A013611
• A094305 → basis A013611
• A038208 → x ↦ x/(x+1)
• A038330 → basis A013611 → x ↦ x/(x+1)
• A178820 → basis A007318 → reverse
• Veronese parent of A098172 → x ↦ x/(x+1)
• A038208 → basis A013611
• A178821 → basis A013609 → basis A013611
• A027466 → basis A013611
• A038221 → basis A013611 → basis A013610
• A134346 → x ↦ x+1
• A038330 → basis A013609 → reverse
• A134059 → basis A007318
• A038329 → x ↦ x+1 → basis A013610
• A178821 → x ↦ x/(x+1)
• A178822 → x ↦ x+1 → reverse
• A028326 → x ↦ x/(x+1)
• basis-map coefficients of A112227 → θ+1 → basis A013609
• A038325 → basis A007318 → D
• A003506 → D → x ↦ x+1
• A038330 → basis A013611 → basis A007318
• A038325 → basis A013609 → basis A007318
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maxi…
• A344565 → x ↦ x/(x+1)
• A094305 → x ↦ x+1 → D
• A136521 → basis A013609
• A178822 → basis A013611 → D
• A135065 → x ↦ x/(x+1)
• A135089 → basis A007318
• A178820 → basis A007318
• A135838 → x ↦ x/(x+1)
• A027465 → D → basis A013609
• A178820 → basis A007318 → D
• A178820 → basis A013611
• A196347 → basis A013611 → D
• A013609 → basis A013610 → D
• A028326 → basis A013609 → basis A013611

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