A038330 → basis A007318 → D

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:            128,              32;
  1:           3072,            1536,             192;
  2:          65536,           49152,           12288,           1024;
  3:        1310720,         1310720,          491520,          81920,           5120;
  4:       25165824,        31457280,        15728640,        3932160,         491520,          24576;
  5:      469762048,       704643072,       440401920,      146800640,       27525120,        2752512,        114688;
  6:     8589934592,     15032385536,     11274289152,     4697620480,     1174405120,      176160768,      14680064,       524288;
  7:   154618822656,    309237645312,    270582939648,   135291469824,    42278584320,     8455716864,    1056964608,     75497472,     2359296;
  8:  2748779069440,   6184752906240,   6184752906240,  3607772528640,  1352914698240,   338228674560,   56371445760,   6039797760,   377487360,   10485760;
  9: 48378511622144, 120946279055360, 136064563937280, 90709709291520, 39685497815040, 11905649344512, 2480343613440, 354334801920, 33218887680, 1845493760, 46137344;

The row polynomials satisfy (n+1)*P(n,x) = (16*n+32+(4*n+8)*x)*P(n-1,x) for n >= 1, with P(0,x) = 128 + 32*x.

P[0, x_] := 128 + 32*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 32 + (4*n + 8)*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] = 128 + 32*x; for(n = 1, N, my(i = n + 1); v[i] = ((16*n + 32 + (4*n + 8)*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_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
            • 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
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A131436 → basis A013611
• basis-map coefficients of A112227 → θ+1 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013611
• A013610 → basis A013611 → x ↦ x/(x+1)
• image under θ+1 → basis A013611
• A027467 → basis A013611 → D
• A027467 → basis A013611 → reverse
• basis-map coefficients of A112227 → basis A013611
• A013611 → D
• A129186 → basis A013611
• A136572 → θ+1 → basis A013611
• basis-map coefficients of A030528, shifted → θ+1 → basis A013611
• basis-map coefficients of A021012 → θ+1 → basis A013611
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013611
• A038303 → basis A007318 → basis A013610
• A136572 → basis A013611 → reverse
• A136521 → basis A013611
• A131431 → D → basis A013611
• A127648 → D → basis A013611
• A132440 → basis A013611
• basis-map coefficients of A026729, shifted → D → basis A013611
• basis-map coefficients of A026729 → D → basis A013611
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
• basis-map coefficients of A026729, shifted → θ+1 → basis A013611
• basis-map coefficients of A026729 → basis A013611 → D
• A131431 → θ+1 → basis A013611
• A136572 → basis A013611 → D
• A136572 → basis A013611
• basis-map coefficients of A112227 → (θ+1)² → basis A013611
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• A038329 → reverse
• basis-map coefficients of A112227 → basis A013611 → reverse
• A038329 → D
• A136572 → D → basis A013611
• A027466 → basis A013609
• A120910 → basis A007318 → basis A007318
• A134309 → basis A013611
• basis-map coefficients of A026729 → basis A013611
• A038330 → basis A007318 → reverse
• A130123 → basis A013611
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013611
• A130123 → (θ+1)² → basis A013611
• image under θ+1 → (θ+1)² → basis A013611
• A038330 → basis A007318
• A013611 → D → D
• A132440 → θ+1 → basis A013611
• A136572 → (θ+1)² → basis A013611
• basis-map coefficients of A021012 → basis A013611
• A131431 → basis A013611
• image under θ+1 → θ+1 → D → basis A013611
• A130123 → basis A013611 → D
• basis-map coefficients of A026729, shifted → basis A013611
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
• basis-map coefficients of A026729 → (θ+1)² → basis A013611
• basis-map coefficients of A030528, shifted → D → basis A013611
• basis-map coefficients of A112227 → basis A013611 → D
• A130123 → θ+1 → basis A013611
• A038231 → D → D
• A130123 → D → basis A013611
• basis-map coefficients of A026729 → basis A013611 → reverse
• A027466 → basis A013609 → D
• A027467 → basis A013611
• A038231 → D

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.