A130850: Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.

OEIS matches ?

• A076726: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A028246: 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 = 1 ?
• A000629: 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 ?
• A130850: 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 ?

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,       1;
  2:      2,       3,       1;
  3:      6,      12,       7,       1;
  4:     24,      60,      50,      15,       1;
  5:    120,     360,     390,     180,      31,       1;
  6:    720,    2520,    3360,    2100,     602,      63,      1;
  7:   5040,   20160,   31920,   25200,   10206,    1932,    127,     1;
  8:  40320,  181440,  332640,  317520,  166824,   46620,   6050,   255,   1;
  9: 362880, 1814400, 3780000, 4233600, 2739240, 1020600, 204630, 18660, 511, 1;

The row polynomials satisfy P(n,x) = (n+n*x)*P(n-1,x) - (x+x^2)*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[(n + n*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
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] = (n + n*x)*v[i - 1] - (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 - 1, v[i])))};
T(9)

Triangle begins:
   1: 1;
   2: 1,   1;
   3: 1,   3,     2;
   4: 1,   7,    12,      6;
   5: 1,  15,    50,     60,      24;
   6: 1,  31,   180,    390,     360,     120;
   7: 1,  63,   602,   2100,    3360,    2520,     720;
   8: 1, 127,  1932,  10206,   25200,   31920,   20160,    5040;
   9: 1, 255,  6050,  46620,  166824,  317520,  332640,  181440,   40320;
  10: 1, 511, 18660, 204630, 1020600, 2739240, 4233600, 3780000, 1814400, 362880;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (n-1+(n-1)*x)*P(n-1,x) - (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 - 1)*x)*P[n - 1, x] - (x + x^2)*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] = (n - 1 + (n - 1)*x)*v[i - 1] - (x + x^2)*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:
  0:      1;
  1:      1,       1;
  2:      2,       3,       1;
  3:      6,      12,       7,       1;
  4:     24,      60,      50,      15,       1;
  5:    120,     360,     390,     180,      31,       1;
  6:    720,    2520,    3360,    2100,     602,      63,      1;
  7:   5040,   20160,   31920,   25200,   10206,    1932,    127,     1;
  8:  40320,  181440,  332640,  317520,  166824,   46620,   6050,   255,   1;
  9: 362880, 1814400, 3780000, 4233600, 2739240, 1020600, 204630, 18660, 511, 1;

The row polynomials satisfy P(n,x) = (n+n*x)*P(n-1,x) - (x+x^2)*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[(n + n*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, 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; for(n = 1, N, my(i = n + 1); v[i] = (n + n*x)*v[i - 1] - (x + x^2)*deriv(v[i - 1], 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 >= 0, 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-222fddff0a5b4a86
• 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-222fddff0a5b4a86, e-afbdda6cf256c7ee
• 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-222fddff0a5b4a86, e-afbdda6cf256c7ee
• Real-rooted. Every row P_n, n >= 0, 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-afbdda6cf256c7ee

Proof chains

2 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• preserver_image reversal, shift 0 e-222fddff0a5b4a86

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check ma_wang_interval e-994b5c04cf13d0cc root: theorem check

      Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

• preserver_image affine, shift 0 e-afbdda6cf256c7ee

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image reversal, shift 1 e-61bc78adfe1eb25a

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-083b494dcfd2f637 root: theorem check

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

        • parent_provers root_domain e-ffc8a9fffd6e96cb root: theorem check

          Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (2)

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

• A334218 → x ↦ x+1
• A028246: Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.

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