A038330 → reverse

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:      16,        96,        144;
  1:      64,       576,       1728,        1728;
  2:     256,      3072,      13824,       27648,        20736;
  3:    1024,     15360,      92160,      276480,       414720,       248832;
  4:    4096,     73728,     552960,     2211840,      4976640,      5971968,       2985984;
  5:   16384,    344064,    3096576,    15482880,     46448640,     83607552,      83607552,      35831808;
  6:   65536,   1572864,   16515072,    99090432,    371589120,    891813888,    1337720832,    1146617856,     429981696;
  7:  262144,   7077888,   84934656,   594542592,   2675441664,   8026324992,   16052649984,   20639121408,   15479341056,    5159780352;
  8: 1048576,  31457280,  424673280,  3397386240,  17836277760,  64210599936,  160526499840,  275188285440,  309586821120,  206391214080,   61917364224;
  9: 4194304, 138412032, 2076180480, 18685624320, 112113745920, 470877732864, 1412633198592, 3027071139840, 4540606709760, 4540606709760, 2724364025856, 743008370688;

The row polynomials satisfy P(n,x) = (4+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 16 + 96*x + 144*x^2.

P[0, x_] := 16 + 96*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(4 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 16 + 96*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (4 + 12*x)*v[i - 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 Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• 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 [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 ?

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

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.