A038330 → D

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:            96,            32;
  1:          1728,          1152,            192;
  2:         27648,         27648,           9216,           1024;
  3:        414720,        552960,         276480,          61440,          5120;
  4:       5971968,       9953280,        6635520,        2211840,        368640,         24576;
  5:      83607552,     167215104,      139345920,       61931520,      15482880,       2064384,       114688;
  6:    1146617856,    2675441664,     2675441664,     1486356480,     495452160,      99090432,     11010048,       524288;
  7:   15479341056,   41278242816,    48157949952,    32105299968,   13377208320,    3567255552,    594542592,     56623104,     2359296;
  8:  206391214080,  619173642240,   825564856320,   642105999360,  321052999680,  107017666560,  23781703680,   3397386240,   283115520,   10485760;
  9: 2724364025856, 9081213419520, 13621820129280, 12108284559360, 7063165992960, 2825266397184, 784796221440, 149484994560, 18685624320, 1384120320, 46137344;

The row polynomials satisfy (n+1)*P(n,x) = (12*n+24+(4*n+8)*x)*P(n-1,x) for n >= 1, with P(0,x) = 96 + 32*x.

P[0, x_] := 96 + 32*x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 24 + (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] = 96 + 32*x; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 24 + (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 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

Related ?

Same fingerprint ?

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