A125273 -id:A125273 - cp's OEIS frontend

A125274 Eigensequence of triangle A078812: a(n) = Sum_{k=0..n-1} A078812(n-1,k)*a(k) for n > 0 with a(0)=1.

1, 1, 3, 10, 42, 210, 1199, 7670, 54224, 418744, 3499781, 31425207, 301324035, 3069644790, 33078375153, 375634524357, 4480492554993, 55971845014528, 730438139266281, 9935106417137098, 140553930403702487

Offset: 0

Paul D. Hanna, Nov 26 2006

nonn

			a(3) = 3*(1) + 4*(1) + 1*(3) = 10;
a(4) = 4*(1) + 10*(1) + 6*(3) + 1*(10) = 42;
a(5) = 5*(1) + 20*(1) + 21*(3) + 8*(10) + 1*(42) = 210.
Triangle A078812(n,k) = binomial(n+k+1, n-k) begins:
  1;
  2,  1;
  3,  4,  1;
  4, 10,  6,  1;
  5, 20, 21,  8,  1;
  6, 35, 56, 36, 10,  1; ...
where g.f. of column k = 1/(1-x)^(2*k+2).

Seiichi Manyama, Table of n, a(n) for n = 0..516
Jeffrey B. Remmel, Consecutive Up-down Patterns in Up-down Permutations, Electron. J. Combin., 21 (2014), #P3.2.

Cf. A078812, A125273 (variant), A351816, A351817, A351818.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n+k, n-k-1] * a[k], {k, 0, n-1}]; Array[a, 20, 0] (* Amiram Eldar, Nov 24 2018 *)

a(n)=if(n==0,1,sum(k=0,n-1, a(k)*binomial(n+k, n-k-1)))

a(n) = Sum_{k=0..n-1} binomial(n+k, n-k-1)*a(k) for n > 0 with a(0)=1.

G.f. satisfies A(x) = 1 + x/(1-x)^2*A(x/(1-x)^2). [Vladimir Kruchinin, Nov 28 2011]

A351813 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - x)^3) / (1 - x).

Original entry on oeis.org

1, 1, 2, 7, 32, 179, 1184, 8977, 76391, 719132, 7405261, 82654011, 992533974, 12744345310, 174073918884, 2518084939316, 38429337167618, 616676966998463, 10374679318111371, 182506045254212184, 3349265281648290030, 63984975864984809787, 1270096455615572678617

Offset: 0

Ilya Gutkovskiy, Feb 19 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..486

Cf. A000110, A125273.

nmax = 22; A[] = 0; Do[A[x] = 1 + x A[x/(1 - x)^3]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 2 k - 1, n - k - 1] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 22}]

a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n+2*k-1,n-k-1) * a(k).

A351814 G.f. A(x) satisfies A(x) = 1 + x * A(x/(1 - x)^4) / (1 - x).

Original entry on oeis.org

1, 1, 2, 8, 42, 272, 2115, 19010, 192760, 2172468, 26896081, 362184998, 5262526484, 81969555736, 1361249430071, 23989460080079, 446832403813788, 8765575657218860, 180544405959236487, 3893718987163468969, 87711985393624557487, 2059264143275898894916

Offset: 0

Ilya Gutkovskiy, Feb 20 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..467

Cf. A000110, A125273, A351813, A351815.

nmax = 21; A[] = 0; Do[A[x] = 1 + x A[x/(1 - x)^4]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 3 k - 1, n - k - 1] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]

a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n+3*k-1,n-k-1) * a(k).

A351815 G.f. A(x) satisfies A(x) = 1 + x * A(x/(1 - x)^5) / (1 - x).

Original entry on oeis.org

1, 1, 2, 9, 53, 386, 3422, 35300, 412084, 5364255, 76952267, 1203835714, 20362911276, 369906504888, 7175947738672, 147944905766929, 3227970924123268, 74264452788294013, 1795825803391367571, 45514495928632484735, 1205981001167335524448, 33331235326744168532151

Offset: 0

Ilya Gutkovskiy, Feb 20 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..452

Cf. A000110, A125273, A351813, A351814.

nmax = 21; A[] = 0; Do[A[x] = 1 + x A[x/(1 - x)^5]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 4 k - 1, n - k - 1] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]

a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n+4*k-1,n-k-1) * a(k).

A144251 Eigensequence of triangle A054142.

Original entry on oeis.org

1, 1, 2, 6, 24, 122, 758, 5606, 48378, 479532, 5390940, 68022932, 954948752, 14804391270, 251815549396, 4673137101108, 94148342547146, 2050127343000170, 48061939075355080, 1208742383083994580, 32507565146820336836, 932149980847656487522, 28423646163259392354386, 919399182232129554488328

Offset: 0

Gary W. Adamson, Sep 16 2008

nonn

Eigensequence of the reversed triangle (A085478) = A125273.

Eigentriangle A144252 has row sums of A144251 shifted: (1, 2, 6, 24, 122,...) with right border = A144251.

			Triangle A054142 begins:
  1;
  1, 1;
  1, 3, 1;
  1, 5, 6, 1;
  1, 7, 15, 10, 1;
  1, 9, 28, 35, 15, 1;
  ...
a(3) = 6 = 1*1 + 3*1 + 1*2
a(4) = 24 = 1*1 + 5*1 + 6*2 + 1*6

Seiichi Manyama, Table of n, a(n) for n = 0..376

Cf. A054142, A085478, A144252,

A054142(n, k) = binomial(2*n-k, k);
a(n) = if (n==0, 1, sum(k=0, n-1, A054142(n-1,k)*a(k))); \\ too slow
lista(nn) = my(v=vector(nn)); v[1] = 1; for (n=2, nn, v[n] = sum(k=0, n-1, A054142(n-2,k)*v[k+1]);); v; \\ Michel Marcus, Jan 17 2025

a(n) = Sum_{k=0..n-1} A054142(n-1,k)*a(k) for n>0 with a(0)=1.

More terms from Seiichi Manyama, May 31 2022

A144252 Eigentriangle, row sums = A144251 shifted, right border = A144251.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 12, 6, 1, 7, 30, 60, 24, 1, 9, 56, 210, 360, 122, 1, 11, 90, 504, 1680, 2562, 758, 1, 13, 132, 990, 5040, 15372, 21224, 5606, 1, 15, 182, 1716, 11880, 36364, 159180, 201816, 47378, 1, 17, 240, 2730, 24024, 157014, 700392, 1849980, 2177010, 479532

Offset: 0

Gary W. Adamson, Sep 16 2008

Right border = A144251: (1, 1, 2, 6, 24, 122, 758,...) with row sums = the same sequence shifted. Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
  1;
  1, 1;
  1, 3, 2;
  1, 5, 12, 6;
  1, 7, 30, 60, 24;
  1, 9, 56, 210, 360, 122;
  1, 11, 90, 504, 1680, 2562, 758;
  1, 13, 132, 990, 5040, 15372, 21224, 5606;
  ...
The triangle is generated from A054142 and its own eigensequence, A144251.
Triangle A054142 =
  1;
  1, 1;
  1, 3, 1;
  1, 5, 6, 1;
  1, 7, 15, 10, 1;
  ...
The eigensequence of A054142 = A144251: (1, 1, 2, 6, 24, 122, 758, 5606,...);
Example: row 3 of A144252 = (1, 5, 12, 6) = termwise products of (1, 5, 6, 1) and (1, 1, 2, 6) = (1*1, 5*1, 6*2, 1*6).

Cf. A144251, A054142, A125273, A085478

A054142(n, k) = binomial(2*n-k, k);
V144251(nn) = my(v=vector(nn)); v[1] = 1; for (n=2, nn, v[n] = sum(k=0, n-1, A054142(n-2,k)*v[k+1]);); v;
row(n) = my(v=V144251(n+1)); vector(n+1, k, A054142(n,k-1) * v[k]); \\ Michel Marcus, Jan 18 2025

Eigentriangle by rows, T(n,k) = A054142(n,k) * A144251(k); were A144251 = the eigensequence of triangle A054142.

More terms from Michel Marcus, Jan 18 2025

A132427 Triangle, read by rows of 2n+1 terms, where T(n,k) = T(n,k-1) + T(n-1,k-2) for n>0, 10 and T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 6, 6, 6, 8, 10, 13, 17, 23, 23, 23, 29, 35, 43, 53, 66, 83, 106, 106, 106, 129, 152, 181, 216, 259, 312, 378, 461, 567, 567, 567, 673, 779, 908, 1060, 1241, 1457, 1716, 2028, 2406, 2867, 3434, 3434, 3434, 4001, 4568, 5241, 6020, 6928, 7988

Offset: 0

Paul D. Hanna, Aug 21 2007

Column 0 equals (essentially) column 1 and the rightmost border.

			Triangle begins:
1;
1, 1, 2;
2, 2, 3, 4, 6;
6, 6, 8, 10, 13, 17, 23;
23, 23, 29, 35, 43, 53, 66, 83, 106;
106, 106, 129, 152, 181, 216, 259, 312, 378, 461, 567;
567, 567, 673, 779, 908, 1060, 1241, 1457, 1716, 2028, 2406, 2867, 3434; ...

Cf. A125273; A132428 (central terms).

t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, k-2]; t[n_, 0] := t[n, 0] = t[n-1, 2n-2]; t[n_, 1] := t[n, 0]; t[0, 0] = 1; Flatten[ Table[t[n, k], {n, 0, 7}, {k, 0, 2 n}]] (* Jean-François Alcover, Jun 18 2012 *)

PARI
```
T(n,k)=local(A=[1]);if(2*n
				
```

The g.f. of column 0 (A125273) satisfies: G(x) = 1 + x*G( x/(1-x)^2 ) / (1-x).

The central terms (A132428) are the inverse binomial transform of A125273 (offset 1).

A132428 Central terms of triangle A132427.

Original entry on oeis.org

1, 1, 3, 10, 43, 216, 1241, 7988, 56763, 440254, 3693728, 33281359, 320112326, 3270177860, 35329070470, 402128329243, 4806784533967, 60166803598106, 786622663286330, 10717555856584617, 151864784070048105

Offset: 0

Paul D. Hanna, Aug 21 2007

Cf. A125273; A132427.

a(n):=if n=0 then 1  else sum(sum(binomial(i-1,k-1)*binomial(i,n-i),i,k,n)*a(k-1),k,1,n); /* Vladimir Kruchinin, May 02 2012 */

{a(n)=local(G=1+x+x*O(x^n));for(i=0,n,G=1+x*subst(G,x,x/(1-x)^2)/(1-x)); polcoeff(subst((G-1)/x,x,x/(1+x))/(1+x),n)}

Binomial transform equals A125273.

a(n)=sum(k=1..n, sum(i=k..n, binomial(i-1,k-1)*binomial(i,n-i))*a(k-1)), n>0,a(0)=1. [Vladimir Kruchinin, May 02 2012]

A144250 Eigentriangle, row sums = A125275, shifted.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 30, 42, 23, 1, 15, 70, 168, 207, 106, 1, 21, 140, 504, 1035, 1166, 567, 1, 28, 252, 1260, 3795, 6996, 7371, 3434

Offset: 0

Gary W. Adamson, Sep 16 2008

Row sums = A125273 shifted. A125273 = the eigensequence of triangle A085478.

Right border = A125273: (1, 1, 2, 6, 23, 106, 567, 3434,...). Sum of n-th row terms = rightmost term in next row.

			First few rows of the triangle =
1;
1, 1;
1, 3, 2;
1, 6, 10, 6;
1, 10, 30, 42, 23;
1, 15, 70, 168, 207, 106;
1, 21, 140, 504, 1035, 1166, 567;
...
Row 4 = (1, 10, 30, 42, 23) = termwise products of (1, 10, 15, 7, 1) and (1, 1, 2, 6, 23) = (1*1, 10*1, 15*2, 7*6, 1*23); where (1, 10, 15, 7, 1) = row 4 of triangle A085478. Q

Cf. A125273, A085478.

Triangle read by rows, T(n,k) = A085478(n,k) * A125273(k).

As infinite lower triangular matrices, A144250 = A085478 * (A125275 * 0^(n-k); where (A125275 * 0^(n-k)) = an infinite lower triangular matrix with A125275: (1, 1, 2, 6, 23, 106, 567, 3434,...) as the main diagonal and the rest zeros.

Corrected definition: Eigentriangle, row sums = A125273, shifted. - Gary W. Adamson, Nov 05 2008

A244888 Triangle of coefficients arising in study of up-down permutations.

Original entry on oeis.org

1, -2, 1, 6, -6, 1, -23, 36, -15, 1, 106, -229, 160, -37, 1, -567, 1574, -1566, 650, -93, 1, 3434, -11706, 15248, -9310, 2572, -238, 1, -23137, 93831, -151933, 123814, -52136, 10175, -616, 1

Offset: 1

N. J. A. Sloane, Jul 12 2014

See Remmel (2014) for precise definition.

			Triangle begins:
       1;
      -2,      1;
       6,     -6,       1;
     -23,     36,     -15,      1;
     106,   -229,     160,    -37,      1;
    -567,   1574,   -1566,    650,    -93,     1;
    3434, -11706,   15248,  -9310,   2572,  -238,    1;
  -23137,  93831, -151933, 123814, -52136, 10175, -616, 1;
  ...

Jeffrey B. Remmel, Consecutive Up-down Patterns in Up-down Permutations, Electron. J. Combin., 21 (2014), #P3.2. See p. 25.

First two columns are A125273 and A244889.

cp's OEIS Frontend

A125274 Eigensequence of triangle A078812: a(n) = Sum_{k=0..n-1} A078812(n-1,k)*a(k) for n > 0 with a(0)=1.

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A351813 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - x)^3) / (1 - x).

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A351814 G.f. A(x) satisfies A(x) = 1 + x * A(x/(1 - x)^4) / (1 - x).

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A351815 G.f. A(x) satisfies A(x) = 1 + x * A(x/(1 - x)^5) / (1 - x).

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A144251 Eigensequence of triangle A054142.

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A144252 Eigentriangle, row sums = A144251 shifted, right border = A144251.

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A132427 Triangle, read by rows of 2n+1 terms, where T(n,k) = T(n,k-1) + T(n-1,k-2) for n>0, 10 and T(0,0)=1.

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A132428 Central terms of triangle A132427.

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Maxima