A134309 → basis A013610

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Submission text ?

Triangle begins:
  0:   1;
  1:   1,    3;
  2:   2,   12,    18;
  3:   4,   36,   108,    108;
  4:   8,   96,   432,    864,     648;
  5:  16,  240,  1440,   4320,    6480,    3888;
  6:  32,  576,  4320,  17280,   38880,   46656,    23328;
  7:  64, 1344, 12096,  60480,  181440,  326592,   326592,   139968;
  8: 128, 3072, 32256, 193536,  725760, 1741824,  2612736,  2239488,   839808;
  9: 256, 6912, 82944, 580608, 2612736, 7838208, 15676416, 20155392, 15116544, 5038848;

The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(6*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.

P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (6*n - 6)*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] = 1; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (6*n - 6)*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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).?
    • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.