A038330 → basis A013611 → (θ+1)²

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Triangle begins:
  0:            256,            2048,             2304;
  1:           4096,           49152,           110592,             65536;
  2:          65536,         1048576,          3538944,           4194304,            1638400;
  3:        1048576,        20971520,         94371840,         167772160,          131072000,           37748736;
  4:       16777216,       402653184,       2264924160,        5368709120,         6291456000,         3623878656,          822083584;
  5:      268435456,      7516192768,      50734301184,      150323855360,       234881024000,       202937204736,        92073361408,        17179869184;
  6:     4294967296,    137438953472,    1082331758592,     3848290697216,      7516192768000,      8658654068736,      5892695130112,      2199023255552,       347892350976;
  7:    68719476736,   2473901162496,   22265110462464,    92358976733184,    216466351718400,    311711546474496,    282849366245376,    158329674399744,     50096498540544,     6871947673600;
  8:  1099511627776,  43980465111040,  445302209249280,  2111062325329920,   5772436045824000,   9974769487183872,  11313974649815040,   8444249301319680,   4007719883243520,  1099511627776000,   133040906960896;
  9: 17592186044416, 774056185954304, 8708132091985920, 46443371157258240, 145135534866432000, 292593238290726912, 398251907673489408, 371546969258065920, 235119566483619840, 96757023244288000, 23415199625117696, 2533274790395904;

The row polynomials satisfy (1+(3*n+5)*x+(n^2+4*n+4)*x^2)*P(n,x) = (16+(48*n+144)*x+(16*n^2+144*n+272)*x^2+(16*n^2+96*n+144)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 2048*x + 2304*x^2.

P[0, x_] := 256 + 2048*x + 2304*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((16 + (48*n + 144)*x + (16*n^2 + 144*n + 272)*x^2 + (16*n^2 + 96*n + 144)*x^3)*P[n - 1, x])/(1 + (3*n + 5)*x + (n^2 + 4*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 256 + 2048*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((16 + (48*n + 144)*x + (16*n^2 + 144*n + 272)*x^2 + (16*n^2 + 96*n + 144)*x^3)*v[i - 1])/(1 + (3*n + 5)*x + (n^2 + 4*n + 4)*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 whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
            • 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
            • 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
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A028326 → D → (θ+1)²
• A120910 → basis A013610 → (θ+1)²
• A125092 → reverse → D
• A038221 → (θ+1)²
• A085881 → basis A013609 → (θ+1)²
• A038325 → (θ+1)²
• basis-map coefficients of A112227 → x ↦ x+1 → (θ+1)²
• A038338 → basis A013609 → (θ+1)²
• A028326 → (θ+1)²
• A038338 → (θ+1)²
• A178820 → basis A013609 → (θ+1)²
• A156992 → (θ+1)²
• basis-map coefficients of A113214, shifted → basis A013609 → (θ+1)²
• A196347 → basis A013609 → (θ+1)²
• A376827 → (θ+1)²
• A085880 → (θ+1)²
• A028326 → basis A013609 → (θ+1)²
• A125092 → reverse
• A196347 → (θ+1)²
• A085881 → (θ+1)²
• A038221 → basis A013609 → (θ+1)²
• A038325 → basis A013609 → (θ+1)²
• A124860 → (θ+1)²
• A003506 → (θ+1)²
• basis-map coefficients of A127647 → (θ+1)²
• A356546 → basis A013609 → (θ+1)²
• A178822 → basis A013609 → (θ+1)²
• A131431 → x ↦ x+1 → (θ+1)²
• A027465 → basis A013611 → (θ+1)²
• A085880 → D → (θ+1)²
• A178822 → (θ+1)²
• A344565 → (θ+1)²
• A178821 → (θ+1)²
• basis-map coefficients of A026729, shifted → basis A007318 → (θ+1)²
• A135838 → (θ+1)²
• A134346 → (θ+1)²
• basis-map coefficients of A113214, shifted → D → (θ+1)²
• A178821 → basis A013609 → (θ+1)²
• basis-map coefficients of A113214, shifted → (θ+1)²
• A135065 → (θ+1)²
• A085880 → basis A013609 → (θ+1)²
• A356546 → (θ+1)²
• A178820 → (θ+1)²
• A003506 → D → (θ+1)²
• basis-map coefficients of A026729 → basis A007318 → (θ+1)²
• A156992 → basis A013609 → (θ+1)²
• A038208 → D → (θ+1)²
• A038208 → (θ+1)²
• A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).

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.