A008277 → x ↦ x+1

OEIS matches ?

• A126350: reversed triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 1 ?
• A137597: 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 ?
• A035009: 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,      1;
   3:      5,      5,      1;
   4:     15,     22,      9,      1;
   5:     52,     99,     61,     14,      1;
   6:    203,    471,    385,    135,     20,     1;
   7:    877,   2386,   2416,   1140,    260,    27,     1;
   8:   4140,  12867,  15470,   9156,   2835,   455,    35,    1;
   9:  21147,  73681, 102215,  72590,  28441,  6230,   742,   44,  1;
  10: 115975, 446620, 700311, 578929, 274386, 76503, 12474, 1146, 54, 1;

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (1+x)*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[(2 + x)*P[n - 1, x] + (1 + x)*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] = (2 + x)*v[i - 1] + (1 + x)*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: 1,  2;
   3: 1,  5,    5;
   4: 1,  9,   22,    15;
   5: 1, 14,   61,    99,    52;
   6: 1, 20,  135,   385,   471,    203;
   7: 1, 27,  260,  1140,  2416,   2386,    877;
   8: 1, 35,  455,  2835,  9156,  15470,  12867,   4140;
   9: 1, 44,  742,  6230, 28441,  72590, 102215,  73681,  21147;
  10: 1, 54, 1146, 12474, 76503, 274386, 578929, 700311, 446620, 115975;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+x)*P(n-1,x) + (1+x)*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[(2 + x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x]];
Table[Reverse[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] = (2 + x)*v[i - 1] + (1 + x)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); 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,     22,      9,      1;
   5:     52,     99,     61,     14,      1;
   6:    203,    471,    385,    135,     20,     1;
   7:    877,   2386,   2416,   1140,    260,    27,     1;
   8:   4140,  12867,  15470,   9156,   2835,   455,    35,    1;
   9:  21147,  73681, 102215,  72590,  28441,  6230,   742,   44,  1;
  10: 115975, 446620, 700311, 578929, 274386, 76503, 12474, 1146, 54, 1;

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (1+x)*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[(2 + x)*P[n - 1, x] + (1 + x)*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] = (2 + x)*v[i - 1] + (1 + x)*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-871f3cfc100895ba
• 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-871f3cfc100895ba
• 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-871f3cfc100895ba
• 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-871f3cfc100895ba

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-871f3cfc100895ba

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= 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-995393ce4e49a8d6 root: theorem check

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

    • basis_map A008277 e-c52960bc5e97caa0

      Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

      • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-6e85c0971e84e272 root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image mobius, shift 0 e-ad67990681a86223

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers periodic_product e-6e85c0971e84e272 (expanded above)
    • preserver_image reversal, shift 0 e-d527982dc7c0aa3b

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= 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-8e580efc86a38a12 root: theorem check

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

        • preserver_image reversal, shift 0 e-3256729839b26765

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= 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-995393ce4e49a8d6 (expanded above)
            • ↵ basis_map A008277 e-c52960bc5e97caa0 (expanded above)

Related sequences ?

Same fingerprint (3)

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

• A008278 → x ↦ x/(x+1)
• A137597: Triangle read by rows: A008277 * A007318.
• A007318 → basis A049020 → reverse

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
2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741