A078812 → x ↦ x/(x+1)

OEIS matches ?

• A060921: reversed triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A002450: 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,   3;
   3:  3,  10,    8;
   4:  4,  22,   38,    21;
   5:  5,  40,  111,   130,    55;
   6:  6,  65,  256,   474,   420,   144;
   7:  7,  98,  511,  1324,  1836,  1308,   377;
   8:  8, 140,  924,  3130,  6020,  6666,  3970,   987;
   9:  9, 192, 1554,  6588, 16435, 25088, 23109, 11822,  2584;
  10: 10, 255, 2472, 12720, 39430, 77645, 98160, 77378, 34690, 6765;

The row polynomials satisfy (2*n-2+(4*n-4)*x+(2*n-2)*x^2)*P(n,x) = (2*n+(3*n+8)*x-(5*n-26)*x^2-(11*n-28)*x^3-(5*n-10)*x^4)*P(n-1,x) + (4*x+17*x^2+27*x^3+19*x^4+5*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[((2*n + (3*n + 8)*x - (5*n - 26)*x^2 - (11*n - 28)*x^3 - (5*n - 10)*x^4)*P[n - 1, x] + (4*x + 17*x^2 + 27*x^3 + 19*x^4 + 5*x^5)*D[P[n - 1, x], x])/(2*n - 2 + (4*n - 4)*x + (2*n - 2)*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] = ((2*n + (3*n + 8)*x - (5*n - 26)*x^2 - (11*n - 28)*x^3 - (5*n - 10)*x^4)*v[i - 1] + (4*x + 17*x^2 + 27*x^3 + 19*x^4 + 5*x^5)*deriv(v[i - 1], x))/(2*n - 2 + (4*n - 4)*x + (2*n - 2)*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:    8,    10,     3;
  3:   21,    38,    22,     4;
  4:   55,   130,   111,    40,     5;
  5:  144,   420,   474,   256,    65,     6;
  6:  377,  1308,  1836,  1324,   511,    98,     7;
  7:  987,  3970,  6666,  6020,  3130,   924,   140,    8;
  8: 2584, 11822, 23109, 25088, 16435,  6588,  1554,  192,   9;
  9: 6765, 34690, 77378, 98160, 77645, 39430, 12720, 2472, 255, 10;

Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n+4*n*x+2*n*x^2)*P(n,x) = (2*n+2+(3*n+11)*x-(5*n-21)*x^2-(11*n-17)*x^3-(5*n-5)*x^4)*P(n-1,x) + (4*x+17*x^2+27*x^3+19*x^4+5*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[((2*n + 2 + (3*n + 11)*x - (5*n - 21)*x^2 - (11*n - 17)*x^3 - (5*n - 5)*x^4)*P[n - 1, x] + (4*x + 17*x^2 + 27*x^3 + 19*x^4 + 5*x^5)*D[P[n - 1, x], x])/(2*n + 4*n*x + 2*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] = ((2*n + 2 + (3*n + 11)*x - (5*n - 21)*x^2 - (11*n - 17)*x^3 - (5*n - 5)*x^4)*v[i - 1] + (4*x + 17*x^2 + 27*x^3 + 19*x^4 + 5*x^5)*deriv(v[i - 1], x))/(2*n + 4*n*x + 2*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-c92b2c850587f23b
• 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-c92b2c850587f23b
• 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-c92b2c850587f23b

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-c92b2c850587f23b

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0)
    • theorem_check three_term_recurrence e-5aa1c8cf88fab58c 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.

• A172431 → basis A007318

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