A104684 → x ↦ x/(x+1)

OEIS matches ?

• A376467: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A006442: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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,      3;
   3:     6,     18,      13;
   4:    20,     90,     132,       63;
   5:    70,    420,     930,      900,      321;
   6:   252,   1890,    5600,     8190,     5910,     1683;
   7:   924,   8316,   30870,    60480,    65940,    37926,     8989;
   8:  3432,  36036,  160776,   395010,   577080,   501228,   239624,    48639;
   9: 12870, 154440,  804804,  2378376,  4358970,  5072760,  3660300,  1497096,  265729;
  10: 48620, 656370, 3912480, 13513500, 29801772, 43513470, 42060480, 25951860, 9274410, 1462563;

The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = (4*n-6+(17*n-27)*x+(26*n-44)*x^2+(17*n-31)*x^3+(4*n-8)*x^4)*P(n-1,x) - (2*x+10*x^2+18*x^3+14*x^4+4*x^5)*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[((4*n - 6 + (17*n - 27)*x + (26*n - 44)*x^2 + (17*n - 31)*x^3 + (4*n - 8)*x^4)*P[n - 1, x] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)]];
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] = ((4*n - 6 + (17*n - 27)*x + (26*n - 44)*x^2 + (17*n - 31)*x^3 + (4*n - 8)*x^4)*v[i - 1] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)); 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:       3,       2;
  2:      13,      18,        6;
  3:      63,     132,       90,       20;
  4:     321,     900,      930,      420,       70;
  5:    1683,    5910,     8190,     5600,     1890,      252;
  6:    8989,   37926,    65940,    60480,    30870,     8316,      924;
  7:   48639,  239624,   501228,   577080,   395010,   160776,    36036,    3432;
  8:  265729, 1497096,  3660300,  5072760,  4358970,  2378376,   804804,  154440,  12870;
  9: 1462563, 9274410, 25951860, 42060480, 43513470, 29801772, 13513500, 3912480, 656370, 48620;

Row n lists the coefficients of P(n,x) in descending powers of x, where (n+2*n*x+n*x^2)*P(n,x) = (4*n-2+(17*n-10)*x+(26*n-18)*x^2+(17*n-14)*x^3+(4*n-4)*x^4)*P(n-1,x) - (2*x+10*x^2+18*x^3+14*x^4+4*x^5)*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[((4*n - 2 + (17*n - 10)*x + (26*n - 18)*x^2 + (17*n - 14)*x^3 + (4*n - 4)*x^4)*P[n - 1, x] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*x^2)]];
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] = ((4*n - 2 + (17*n - 10)*x + (26*n - 18)*x^2 + (17*n - 14)*x^3 + (4*n - 4)*x^4)*v[i - 1] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*x^2)); 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 [-1, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-22fed0d4659f5966
• 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-22fed0d4659f5966
• 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-22fed0d4659f5966

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 mobius, shift 0 e-22fed0d4659f5966

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-74c20febd3423e2e root: theorem check

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

    • preserver_image reversal, shift 0 e-5047c71cb920acb6

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-18de01688f1c88b0 root: theorem check

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

        • theorem_check three_term_recurrence e-93f30859d3148f4f root: theorem check

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

        • preserver_image reversal, shift 0 e-00d47151fcd2720e

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,…
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check three_term_recurrence e-74c20febd3423e2e (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A063007 → x ↦ x+1

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023