A321331: Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).

OEIS matches ?

• A005493: 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 ?
• A138378: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
• A321331: triangle, 66 terms agree, 11 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 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,    2;
   3: 1,    6,     3;
   4: 1,   14,    18,      4;
   5: 1,   30,    75,     40,      5;
   6: 1,   62,   270,    260,     75,      6;
   7: 1,  126,   903,   1400,    700,    126,     7;
   8: 1,  254,  2898,   6804,   5250,   1596,   196,    8;
   9: 1,  510,  9075,  31080,  34755,  15876,  3234,  288,   9;
  10: 1, 1022, 27990, 136420, 212625, 136962, 41160, 6000, 405, 10;

The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) + x*d/dx P(n-1,x) - x^2*P(n-2,x) - x^2*d/dx P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.

P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x] - x^2*P[n - 2, x] - x^2*D[P[n - 2, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = (1 + 2*x)*v[i - 1] + x*deriv(v[i - 1], x) - x^2*v[i - 2] - x^2*deriv(v[i - 2], 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:
  0: 1;
  1: 1,    2;
  2: 1,    6,     3;
  3: 1,   14,    18,      4;
  4: 1,   30,    75,     40,      5;
  5: 1,   62,   270,    260,     75,      6;
  6: 1,  126,   903,   1400,    700,    126,     7;
  7: 1,  254,  2898,   6804,   5250,   1596,   196,    8;
  8: 1,  510,  9075,  31080,  34755,  15876,  3234,  288,   9;
  9: 1, 1022, 27990, 136420, 212625, 136962, 41160, 6000, 405, 10;

The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) + x*d/dx P(n-1,x) - x^2*P(n-2,x) - x^2*d/dx P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x] - x^2*P[n - 2, x] - x^2*D[P[n - 2, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 1] + x*deriv(v[i - 1], x) - x^2*v[i - 2] - x^2*deriv(v[i - 2], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #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 (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-907e3c6c6275331d
• 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-907e3c6c6275331d
• 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-907e3c6c6275331d
• 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-907e3c6c6275331d

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 euler_polynomial, shift 0 e-907e3c6c6275331d

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] 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)

Used in 6 proof steps

• e-46e01729cbc6373b preserver_image (affine, shift 0) → derivative of Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,.…
• e-6878e8dc6c11eb74 preserver_image (reversal, shift 0) → reversal of Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 …
• e-888961e617a41267 preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >=…
• e-8b35b367ffdd04e7 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >=…
• e-b376e17507db55c7 preserver_image (derivative_polynomial, shift 0) → derivative of Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S…
• e-b4b600b457a94a2b preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of Triangle of Stirling numbers of the second kind, S2(n,k), n >= …

Related sequences ?

Same fingerprint (2)

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

• A008277 → θ+1
• A008277 → θ+1 → reverse

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.