A136521 → basis A013611

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Triangle begins:
  0: 1;
  1: 2,  8;
  2: 2, 16,   32;
  3: 2, 24,   96,   128;
  4: 2, 32,  192,   512,   512;
  5: 2, 40,  320,  1280,  2560,   2048;
  6: 2, 48,  480,  2560,  7680,  12288,   8192;
  7: 2, 56,  672,  4480, 17920,  43008,  57344,   32768;
  8: 2, 64,  896,  7168, 35840, 114688, 229376,  262144,  131072;
  9: 2, 72, 1152, 10752, 64512, 258048, 688128, 1179648, 1179648, 524288;

The row polynomials satisfy (n-1)*P(n,x) = (n-1+(4*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.

P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (4*n - 4)*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] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (4*n - 4)*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: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.?
    • 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 → basis A013611 → x ↦ x/(x+1)
• A136572 → (θ+1)² → basis A013611
• A027467 → basis A013611
• basis-map coefficients of A112227 → (θ+1)² → basis A013611
• basis-map coefficients of A026729, shifted → D → basis A013611
• A038231 → D
• A136572 → θ+1 → basis A013611
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013611
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013611
• basis-map coefficients of A112227 → basis A013611
• A127648 → D → basis A013611
• A136572 → basis A013611 → D
• A038329 → reverse
• basis-map coefficients of A026729 → basis A013611 → reverse
• A130123 → (θ+1)² → basis A013611
• A013611 → D → D
• A038329 → D
• A038231 → D → D
• A038330 → basis A007318
• A027467 → basis A013611 → D
• basis-map coefficients of A026729, shifted → basis A013611
• basis-map coefficients of A021012 → basis A013611
• A038330 → basis A007318 → D
• basis-map coefficients of A112227 → basis A013611 → D
• basis-map coefficients of A112227 → basis A013611 → reverse
• basis-map coefficients of A026729 → (θ+1)² → basis A013611
• A038303 → basis A007318 → basis A013610
• A027467 → basis A013611 → reverse
• A131436 → basis A013611
• basis-map coefficients of A026729 → D → basis A013611
• A130123 → D → basis A013611
• image under θ+1 → (θ+1)² → basis A013611
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• A132440 → basis A013611
• A038330 → basis A007318 → reverse
• A027466 → basis A013609 → D
• A131431 → θ+1 → basis A013611
• basis-map coefficients of A021012 → θ+1 → basis A013611
• A136572 → D → basis A013611
• A136572 → basis A013611
• basis-map coefficients of A026729 → basis A013611 → D
• image under θ+1 → θ+1 → D → basis A013611
• A130123 → θ+1 → basis A013611
• basis-map coefficients of A026729 → basis A013611
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
• A131431 → basis A013611
• basis-map coefficients of A112227 → θ+1 → basis A013611
• basis-map coefficients of A026729, shifted → θ+1 → basis A013611
• basis-map coefficients of A030528, shifted → θ+1 → basis A013611
• image under θ+1 → basis A013611
• A136572 → basis A013611 → reverse
• A130123 → basis A013611 → D
• A027466 → basis A013609
• A129186 → basis A013611
• A130123 → basis A013611
• A132440 → θ+1 → basis A013611
• A013611 → D
• basis-map coefficients of A030528, shifted → basis A013611
• A131431 → D → basis A013611
• A134309 → basis A013611
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
• A120910 → basis A007318 → basis A007318
• basis-map coefficients of A030528, shifted → D → basis A013611

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.