A172431 → x ↦ x/(x+1)

OEIS matches ?

• A195503: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
• A125662: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A010905: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A001353: 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:
  0: 1;
  1: 1,  3;
  2: 1,  6,   8;
  3: 1,  9,  25,   21;
  4: 1, 12,  51,   90,   55;
  5: 1, 15,  86,  234,  300,   144;
  6: 1, 18, 130,  480,  951,   954,   377;
  7: 1, 21, 183,  855, 2305,  3573,  2939,   987;
  8: 1, 24, 245, 1386, 4740, 10008, 12707,  8850,  2584;
  9: 1, 27, 316, 2100, 8715, 23373, 40426, 43398, 26195, 6765;

The row polynomials satisfy (2*n+4*n*x+2*n*x^2)*P(n,x) = (2*n+(13*n-3)*x+(25*n-11)*x^2+(19*n-13)*x^3+(5*n-5)*x^4)*P(n-1,x) - (x+8*x^2+18*x^3+16*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 + (13*n - 3)*x + (25*n - 11)*x^2 + (19*n - 13)*x^3 + (5*n - 5)*x^4)*P[n - 1, x] - (x + 8*x^2 + 18*x^3 + 16*x^4 + 5*x^5)*D[P[n - 1, x], x])/(2*n + 4*n*x + 2*n*x^2)]];
Table[CoefficientList[P[n, x], x], {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 + (13*n - 3)*x + (25*n - 11)*x^2 + (19*n - 13)*x^3 + (5*n - 5)*x^4)*v[i - 1] - (x + 8*x^2 + 18*x^3 + 16*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};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Triangle begins:
  0:    1;
  1:    3,     1;
  2:    8,     6,     1;
  3:   21,    25,     9,     1;
  4:   55,    90,    51,    12,     1;
  5:  144,   300,   234,    86,    15,    1;
  6:  377,   954,   951,   480,   130,   18,    1;
  7:  987,  2939,  3573,  2305,   855,  183,   21,   1;
  8: 2584,  8850, 12707, 10008,  4740, 1386,  245,  24,  1;
  9: 6765, 26195, 43398, 40426, 23373, 8715, 2100, 316, 27, 1;

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+(13*n-3)*x+(25*n-11)*x^2+(19*n-13)*x^3+(5*n-5)*x^4)*P(n-1,x) - (x+8*x^2+18*x^3+16*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 + (13*n - 3)*x + (25*n - 11)*x^2 + (19*n - 13)*x^3 + (5*n - 5)*x^4)*P[n - 1, x] - (x + 8*x^2 + 18*x^3 + 16*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 + (13*n - 3)*x + (25*n - 11)*x^2 + (19*n - 13)*x^3 + (5*n - 5)*x^4)*v[i - 1] - (x + 8*x^2 + 18*x^3 + 16*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 >= 0, 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-83ce870cdc6237f7
• Coefficients of one sign. Every nonzero row, n >= 0: 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-83ce870cdc6237f7
• Hadamard kernel (Malo). Rows P_n, n >= 0, 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-83ce870cdc6237f7

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-83ce870cdc6237f7

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Even row Pascal-square read by antidiagonals.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-83282836b5ac538e root: theorem check

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

Related sequences ?

Candidate equalities (1)

Definitions that agree on the stored rows without an all-n certificate.

• A125662 → reverse (15 rows compared; no all-index equality certificate was found)

Same fingerprint (2)

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

• A125662: A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
• A078812 → 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