A178820 → basis A013610 → basis A013609

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Triangle begins:
  0:         1;
  1:        20,         24;
  2:       250,        600,         360;
  3:      2500,       9000,       10800,        4320;
  4:     21875,     105000,      189000,      151200,        45360;
  5:    175000,    1050000,     2520000,     3024000,      1814400,       435456;
  6:   1312500,    9450000,    28350000,    45360000,     40824000,     19595520,      3919104;
  7:   9375000,   78750000,   283500000,   567000000,    680400000,    489888000,    195955200,    33592320;
  8:  64453125,  618750000,  2598750000,  6237000000,   9355500000,   8981280000,   5388768000,  1847577600,   277136640;
  9: 429687500, 4640625000, 22275000000, 62370000000, 112266000000, 134719200000, 107775360000, 55427328000, 16628198400, 2217093120;

The row polynomials satisfy n*P(n,x) = (5*n+15+(6*n+18)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 15 + (6*n + 18)*x)*P[n - 1, 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] = ((5*n + 15 + (6*n + 18)*x)*v[i - 1])/(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 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)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) = C(n+3,3) * C(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) = C(n+3,3) * C(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) = C(n+3,3) * C(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) = C(n+3,3) * C(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_A007318 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle read by rows: T(n,k) = C(n+3,3) * C(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) = C(n+3,3) * C(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 Triangle read by rows: T(n,k) = C(n+3,3) * C(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 Triangle read by rows: T(n,k) = C(n+3,3) * C(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)

Related ?

Same fingerprint ?

• A085880 → basis A013610 → basis A013609
• A038291 → basis A013609 → x ↦ x/(x+1)
• A038271 → basis A013609 → D
• A038334 → basis A013610
• A028326 → basis A013610 → basis A013609
• A038336 → D
• A178822 → basis A013610 → basis A013609
• A038221 → basis A013610 → basis A013609
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
• A038334 → basis A013610 → reverse
• A178821 → basis A013610 → basis A013609
• A038220 → basis A013610
• A038271 → basis A013609
• A156992 → basis A013610 → basis A013609
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
• A013612 → x ↦ x+1 → D
• A038255 → basis A013610 → basis A013611
• A013612 → x ↦ x+1
• A038334 → basis A013610 → D
• A120910 → basis A013609 → basis A013610
• A013612 → x ↦ x+1 → reverse
• A038259 → D
• A038314 → D
• A038338 → basis A013610 → basis A013609
• A038259: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.
• A038220 → basis A013610 → D
• A038329 → x ↦ x+1 → x ↦ x/(x+1)
• A147716 → basis A013610 → x ↦ x/(x+1)
• A196347 → basis A013610 → basis A013609

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.