A134309 → basis A013611

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Triangle begins:
  0:   1;
  1:   1,    4;
  2:   2,   16,     32;
  3:   4,   48,    192,     256;
  4:   8,  128,    768,    2048,    2048;
  5:  16,  320,   2560,   10240,   20480,    16384;
  6:  32,  768,   7680,   40960,  122880,   196608,   131072;
  7:  64, 1792,  21504,  143360,  573440,  1376256,  1835008,   1048576;
  8: 128, 4096,  57344,  458752, 2293760,  7340032, 14680064,  16777216,   8388608;
  9: 256, 9216, 147456, 1376256, 8257536, 33030144, 88080384, 150994944, 150994944, 67108864;

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

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (8*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] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (8*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 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 ?

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