A114656 → θ+1

OEIS matches ?

• A156136: reversed triangle, 50 terms agree, 9 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A047781: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
   1:   1;
   2:   2,     2;
   3:   4,    12,      3;
   4:   8,    48,     36,      4;
   5:  16,   160,    240,     80,      5;
   6:  32,   480,   1200,    800,    150,      6;
   7:  64,  1344,   5040,   5600,   2100,    252,      7;
   8: 128,  3584,  18816,  31360,  19600,   4704,    392,     8;
   9: 256,  9216,  64512, 150528, 141120,  56448,   9408,   576,   9;
  10: 512, 23040, 207360, 645120, 846720, 508032, 141120, 17280, 810, 10;

The row polynomials satisfy (n^2-n)*P(n,x) = (2*n^2-2*n+(3*n^2-5*n+2)*x)*P(n-1,x) + ((4*n-2)*x-(2*n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 2*n + (3*n^2 - 5*n + 2)*x)*P[n - 1, x] + ((4*n - 2)*x - (2*n - 1)*x^2)*D[P[n - 1, x], x])/(n^2 - n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n^2 - 2*n + (3*n^2 - 5*n + 2)*x)*v[i - 1] + ((4*n - 2)*x - (2*n - 1)*x^2)*deriv(v[i - 1], x))/(n^2 - n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Triangle begins:
  0:  1;
  1:  2,   2;
  2:  3,  12,     4;
  3:  4,  36,    48,      8;
  4:  5,  80,   240,    160,     16;
  5:  6, 150,   800,   1200,    480,     32;
  6:  7, 252,  2100,   5600,   5040,   1344,     64;
  7:  8, 392,  4704,  19600,  31360,  18816,   3584,    128;
  8:  9, 576,  9408,  56448, 141120, 150528,  64512,   9216,   256;
  9: 10, 810, 17280, 141120, 508032, 846720, 645120, 207360, 23040, 512;

Row n lists the coefficients of P(n,x) in descending powers of x, where (n^2+n)*P(n,x) = (2*n^2+2*n+(3*n^2+n)*x)*P(n-1,x) + ((4*n+2)*x-(2*n+1)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 2*n + (3*n^2 + n)*x)*P[n - 1, x] + ((4*n + 2)*x - (2*n + 1)*x^2)*D[P[n - 1, x], x])/(n^2 + n)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^2 + 2*n + (3*n^2 + n)*x)*v[i - 1] + ((4*n + 2)*x - (2*n + 1)*x^2)*deriv(v[i - 1], x))/(n^2 + n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-57c7b63013beaaa7
• Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-57c7b63013beaaa7
• Hadamard kernel (Malo). Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-57c7b63013beaaa7
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-57c7b63013beaaa7

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• preserver_image euler_polynomial, shift 0 e-57c7b63013beaaa7

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-e6c0603bd7c28948 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A114655 → θ+1

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023