A122848: Exponential Riordan array (1, x(1+x/2)).

OEIS matches ?

• A100861: reversed triangle, 64 terms agree, 15 full rows row lengths from the atlas (tabf) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 992 terms from n = 0 ?
• A000085: 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 ?
• A122848: triangle, 80 terms agree, 12 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A144299: reversed triangle, 78 terms agree, 12 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: 0, 1;
  2: 0, 1, 1;
  3: 0, 0, 3,  1;
  4: 0, 0, 3,  6,   1;
  5: 0, 0, 0, 15,  10,   1;
  6: 0, 0, 0, 15,  45,  15,    1;
  7: 0, 0, 0,  0, 105, 105,   21,   1;
  8: 0, 0, 0,  0, 105, 420,  210,  28,  1;
  9: 0, 0, 0,  0,   0, 945, 1260, 378, 36, 1;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + (n - 1)*x*v[i - 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: 1;
  2: 1,  1;
  3: 1,  3;
  4: 1,  6,   3;
  5: 1, 10,  15;
  6: 1, 15,  45,   15;
  7: 1, 21, 105,  105;
  8: 1, 28, 210,  420, 105;
  9: 1, 36, 378, 1260, 945;

Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = x*P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + (n - 1)*x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1, 1;
  3: 0, 0, 3,  1;
  4: 0, 0, 3,  6,   1;
  5: 0, 0, 0, 15,  10,   1;
  6: 0, 0, 0, 15,  45,  15,    1;
  7: 0, 0, 0,  0, 105, 105,   21,   1;
  8: 0, 0, 0,  0, 105, 420,  210,  28,  1;
  9: 0, 0, 0,  0,   0, 945, 1260, 378, 36, 1;

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

P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (n - 1)*x*P[n - 2, 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] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + (n - 1)*x*v[i - 2]); 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)

Triangle begins:
  0: 1;
  1: 1,  0;
  2: 1,  1,   0;
  3: 1,  3,   0,    0;
  4: 1,  6,   3,    0,   0;
  5: 1, 10,  15,    0,   0, 0;
  6: 1, 15,  45,   15,   0, 0, 0;
  7: 1, 21, 105,  105,   0, 0, 0, 0;
  8: 1, 28, 210,  420, 105, 0, 0, 0, 0;
  9: 1, 36, 378, 1260, 945, 0, 0, 0, 0, 0;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = x*P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + (n - 1)*x*v[i - 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 (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-01d50252f4aaa594
• 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-01d50252f4aaa594
• 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-01d50252f4aaa594

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 reversal, shift 0 e-01d50252f4aaa594

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • gamma_polynomial scale 1, shift 0 e-037c1a08736a9fd9

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang_derivative_lag e-e7942008ad3e1420 root: theorem check

          Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

        • preserver_image reversal, shift 0 e-7078061f6d32a158

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang_derivative_lag e-e7942008ad3e1420 (expanded above)
    • parent_provers paper_theorem e-8187ad69de0e901a root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image reversal, shift 0 e-a3a653eabafbad6b

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Exponential Riordan array (1, x(1+x/2)).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        no recorded property covers this requirement

Used in 12 proof steps

• e-140f8c82cfd93ce7 basis_map (A008277) → basis_map_A008277 of Exponential Riordan array (1, x(1+x/2)).
• e-632ffe6aba666124 basis_map (A013611) → basis_map_A013611 of Exponential Riordan array (1, x(1+x/2)).
• e-a15a4571c9a83032 basis_map (A013610) → basis_map_A013610 of Exponential Riordan array (1, x(1+x/2)).
• e-b3e711c86f592bbc basis_map (A013609) → basis_map_A013609 of Exponential Riordan array (1, x(1+x/2)).
• e-d0b0c6817e15c3a8 basis_map (A007318) → basis_map_A007318 of Exponential Riordan array (1, x(1+x/2)).
• e-095bd663860ef885 preserver_image (euler_polynomial, shift 0) → theta_plus_one of Exponential Riordan array (1, x(1+x/2)).
• e-38acf07bf4ec9f74 preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of Exponential Riordan array (1, x(1+x/2)).
• e-64733a2a119dbb75 preserver_image (affine, shift 0) → basis_map_A007318 of Exponential Riordan array (1, x(1+x/2)).
• e-65f84d661e37675b preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Exponential Riordan array (1, x(1+x/2)).
• e-a3a653eabafbad6b preserver_image (reversal, shift 0) → Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the compl…
• e-c5e95bdc41953def preserver_image (derivative_polynomial, shift 0) → derivative of Exponential Riordan array (1, x(1+x/2)).
• e-fc862e6fd76dbdc8 preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Exponential Riordan array (1, x(1+x/2)).

Related sequences ?

Same fingerprint (2)

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

• A100861: Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).
• A100861 → D

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