A356546 → basis A013611 → (θ+1)²

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Submission text ?

Triangle begins:
  0:        1;
  1:        4,         32;
  2:       24,        384,         864;
  3:      160,       3840,       17280,        20480;
  4:     1120,      35840,      241920,       573440,        448000;
  5:     8064,     322560,     2903040,     10321920,      16128000,       9289728;
  6:    59136,    2838528,    31933440,    151388160,     354816000,     408748032,     185450496;
  7:   439296,   24600576,   332107776,   1968046080,    6150144000,   10627448832,    9643425792,    3598712832;
  8:  3294720,  210862080,  3321077760,  23616552960,   92252160000,  212548976640,  289302773760,  215922769920,   68319313920;
  9: 24893440, 1792327680, 32261898240, 267654266880, 1254629376000, 3613332602880, 6557529538560, 7341374177280, 4645713346560, 1274544128000;

The row polynomials satisfy (n+(6*n^2-2*n)*x+4*n^3*x^2)*P(n,x) = (8*n-4+(48*n^2+24*n-24)*x+(32*n^3+144*n^2+16*n-48)*x^2+(64*n^3+96*n^2-32)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - 4 + (48*n^2 + 24*n - 24)*x + (32*n^3 + 144*n^2 + 16*n - 48)*x^2 + (64*n^3 + 96*n^2 - 32)*x^3)*P[n - 1, x])/(n + (6*n^2 - 2*n)*x + 4*n^3*x^2)]];
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] = ((8*n - 4 + (48*n^2 + 24*n - 24)*x + (32*n^3 + 144*n^2 + 16*n - 48)*x^2 + (64*n^3 + 96*n^2 - 32)*x^3)*v[i - 1])/(n + (6*n^2 - 2*n)*x + 4*n^3*x^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 Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A013611 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).?
            • ↑ 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)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (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) = RisingFactorial(n + 1, n) / (k! * (n - k)!).?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A013609 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).?
            • Geometric basis map (elementary) polytool ? step ›

              • from Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (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) = RisingFactorial(n + 1, n) / (k! * (n - k)!).?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                • ↑ Iterated real-rootedness preservers (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A085880 → basis A013611 → (θ+1)²
• basis-map coefficients of A026729 → basis A013609 → (θ+1)²
• A120909 → basis A013610 → (θ+1)²
• A196347 → basis A013611 → (θ+1)²
• A085881 → basis A013611 → (θ+1)²
• A178822 → basis A013611 → (θ+1)²
• A156992 → basis A013611 → (θ+1)²
• basis-map coefficients of A026729, shifted → basis A013609 → (θ+1)²
• A038208 → basis A013611 → (θ+1)²
• A013609 → basis A013610 → (θ+1)²
• A136572 → basis A013609 → (θ+1)²
• A130123 → basis A013609 → (θ+1)²
• A120909 → (θ+1)²
• A028326 → basis A013611 → (θ+1)²
• A013609 → (θ+1)²
• A038221 → basis A013611 → (θ+1)²
• A178821 → basis A013611 → (θ+1)²
• basis-map coefficients of A112227 → basis A013609 → (θ+1)²
• A013609 → D → (θ+1)²
• A178820 → basis A013611 → (θ+1)²
• basis-map coefficients of A030528, shifted → basis A013609 → (θ+1)²

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.