A106800 → x ↦ x+1

OEIS matches ?

• A059340: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
• A004211: 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:     1;
   3:     2,     1;
   4:     5,     5,      1;
   5:    15,    23,     10,      1;
   6:    52,   109,     76,     19,      1;
   7:   203,   544,    531,    224,     36,     1;
   8:   877,  2876,   3641,   2204,    631,    69,    1;
   9:  4140, 16113,  25208,  20089,   8471,  1749,  134,   1;
  10: 21147, 95495, 178564, 177631, 100171, 31331, 4838, 263, 1;

The row polynomials satisfy P(n,x) = (n-1+(n-2)*x)*P(n-1,x) - (1+2*x+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[(n - 1 + (n - 2)*x)*P[n - 1, x] - (1 + 2*x + x^2)*D[P[n - 1, x], x]];
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] = (n - 1 + (n - 2)*x)*v[i - 1] - (1 + 2*x + x^2)*deriv(v[i - 1], x)); 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:
   1:      1;
   2:      2,      1;
   3:      5,      5,       1;
   4:     15,     23,      10,       1;
   5:     52,    109,      76,      19,       1;
   6:    203,    544,     531,     224,      36,      1;
   7:    877,   2876,    3641,    2204,     631,     69,      1;
   8:   4140,  16113,   25208,   20089,    8471,   1749,    134,     1;
   9:  21147,  95495,  178564,  177631,  100171,  31331,   4838,   263,   1;
  10: 115975, 597155, 1302451, 1559788, 1109473, 471827, 113807, 13454, 520, 1;

The row polynomials satisfy P(n,x) = (n+(n-1)*x)*P(n-1,x) - (1+2*x+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[(n + (n - 1)*x)*P[n - 1, x] - (1 + 2*x + x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

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

Proved properties

• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-af1b637d986229c4
• 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-af1b637d986229c4
• 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-af1b637d986229c4
• 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-af1b637d986229c4

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 affine, shift 0 e-af1b637d986229c4

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-2ab1dfe057e84c9d root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

Same fingerprint (5)

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

• A106800 → reverse (shift +1) → x ↦ x/(x+1)
• A059340: Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.
• A008277 → x ↦ x/(x+1)
• A008278 → x ↦ x+1 (shift +1) → reverse
• A008278 → x ↦ x+1

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003