A178821 → basis A013611 → x ↦ x/(x+1)

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Submission text ?

Triangle begins:
  0:      1;
  1:     10,      30;
  2:     60,     360,       540;
  3:    280,    2520,      7560,      7560;
  4:   1120,   13440,     60480,    120960,      90720;
  5:   4032,   60480,    362880,   1088640,    1632960,      979776;
  6:  13440,  241920,   1814400,   7257600,   16329600,    19595520,     9797760;
  7:  42240,  887040,   7983360,  39916800,  119750400,   215550720,   215550720,    92378880;
  8: 126720, 3041280,  31933440, 191600640,  718502400,  1724405760,  2586608640,  2217093120,   831409920;
  9: 366080, 9884160, 118609920, 830269440, 3736212480, 11208637440, 22417274880, 28822210560, 21616657920, 7205552640;

The row polynomials satisfy (n+n*x)*P(n,x) = (2*n+8+(8*n+32)*x+(6*n+24)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 8 + (8*n + 32)*x + (6*n + 24)*x^2)*P[n - 1, x])/(n + n*x)]];
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] = ((2*n + 8 + (8*n + 32)*x + (6*n + 24)*x^2)*v[i - 1])/(n + n*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 [-1, 0]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.?
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Iterated real-rootedness preservers (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Iterated real-rootedness preservers (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Iterated real-rootedness preservers (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.?
            • ↑ Geometric basis map (elementary) (above)
            • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A038330 → D
• basis-map coefficients of A030528, shifted → θ+1 → basis A013610
• A120910 → D → x ↦ x+1
• A027465: Cube of lower triangular normalized binomial matrix.
• A038279 → basis A013610 → reverse
• A027465 → D
• A027465 → D → D
• A038338 → x ↦ x+1 → basis A007318
• A131431 → D → basis A013610
• image under θ+1 → θ+1 → D → basis A013610
• basis-map coefficients of A112227 → basis A013610 → basis A013611
• basis-map coefficients of A026729 → basis A013610
• A196347 → basis A013611 → x ↦ x/(x+1)
• A356546 → basis A013611 → x ↦ x/(x+1)
• A136572 → θ+1 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013611
• A178821 → basis A007318 → x ↦ x+1
• basis-map coefficients of A112227 → θ+1 → basis A013610
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
• A136572 → D → basis A013610
• A134309 → basis A013610
• A038279 → basis A013610 → D
• basis-map coefficients of A026729 → basis A013610 → reverse
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → reverse
• A038330 → basis A013609 → x ↦ x+1
• A120909 → x ↦ x/(x+1)
• A038325 → basis A007318 → basis A007318
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
• basis-map coefficients of A021012 → θ+1 → basis A013610
• A038330 → reverse
• basis-map coefficients of A021012 → basis A013610
• A147716 → basis A013610 → basis A013609
• A127648 → D → basis A013610
• basis-map coefficients of A112227 → basis A013610 → D
• basis-map coefficients of A030528, shifted → basis A013610
• A120909 → basis A013610 → x ↦ x/(x+1)
• A085880 → x ↦ x+1 → x ↦ x+1
• A130123 → D → basis A013610
• basis-map coefficients of A112227 → basis A013610 → reverse
• A132440 → θ+1 → basis A013610
• A038221 → basis A013611 → x ↦ x/(x+1)
• A129186 → basis A013610
• A127648 → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610 → reverse
• A038279 → basis A013610
• A120910 → basis A007318
• A013610 → basis A013611
• basis-map coefficients of A112227 → (θ+1)² → basis A013610
• A135065 → x ↦ x+1 → basis A007318
• basis-map coefficients of A026729 → D → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013611
• A131436 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → D
• A013610 → D → D
• image under θ+1 → basis A013610
• A136521 → basis A013610
• A085880 → basis A013611 → x ↦ x/(x+1)
• A038329 → x ↦ x+1 → basis A013609
• A131431 → θ+1 → basis A013610
• A130123 → basis A013610 → D
• basis-map coefficients of A026729, shifted → basis A013610
• A094305 → x ↦ x+1 → x ↦ x+1
• A013610 → basis A013611 → reverse
• A178820 → basis A013611 → x ↦ x/(x+1)
• A130123 → basis A013610 → basis A013611
• A136572 → (θ+1)² → basis A013610
• basis-map coefficients of A030528, shifted → D → basis A013610
• A028326 → basis A013611 → x ↦ x/(x+1)
• A178820 → basis A007318 → basis A007318
• A131431 → basis A013610
• A156992 → basis A013611 → x ↦ x/(x+1)
• A136572 → basis A013610 → basis A013611
• basis-map coefficients of A026729 → (θ+1)² → basis A013610
• A013610 → basis A013611 → D
• A013609 → basis A013610 → x ↦ x/(x+1)
• A136572 → basis A013610 → D
• A013610 → D → basis A013611
• A156992 → basis A007318 → x ↦ x+1
• A130123 → θ+1 → basis A013610
• basis-map coefficients of A026729, shifted → D → basis A013610
• basis-map coefficients of A026729 → basis A013610 → D
• A038243 → basis A013609 → D
• A196347 → x ↦ x+1 → basis A007318
• A130123 → basis A013610
• A085881 → basis A013611 → x ↦ x/(x+1)
• A136572 → basis A013610
• A013610 → D
• basis-map coefficients of A026729 → basis A013610 → basis A013611
• A120910 → basis A007318 → reverse
• basis-map coefficients of A026729, shifted → θ+1 → basis A013610
• basis-map coefficients of A112227 → basis A013610
• A178822 → x ↦ x+1 → x ↦ x+1
• A038243 → basis A013609
• basis-map coefficients of A026729, shifted → basis A013610 → D
• A130123 → (θ+1)² → basis A013610
• A085881 → basis A007318 → basis A007318
• A028326 → basis A007318 → basis A007318
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013610
• A356546 → x ↦ x+1 → x ↦ x+1
• A178822 → basis A013611 → x ↦ x/(x+1)
• image under θ+1 → (θ+1)² → basis A013610

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.