A035324 -id:A035324 - cp's OEIS frontend

A049375 A convolution triangle of numbers obtained from A034687.

1, 15, 1, 275, 30, 1, 5500, 775, 45, 1, 115500, 19250, 1500, 60, 1, 2502500, 471625, 44625, 2450, 75, 1, 55412500, 11495000, 1254000, 85000, 3625, 90, 1, 1246781250, 279675000, 34093125, 2698875, 143750, 5025, 105, 1, 28398906250, 6802812500

Offset: 1

Wolfdieter Lang

a(n,1) = A034687(n). a(n,m)=: s2(6; n,m), a member of a sequence of unsigned triangles including s2(2; n,m)= A007318(n-1,m-1) (Pascal's triangle). s2(3; n,m)= A035324(n,m), s2(4; n,m)= A035529(n,m), s2(5; n,m)= A048882(n,m).

			{1}; {15,1}; {275,30,1}; {5500,775,45,1}; ...

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Cf. A039746.

a[n_, m_] := Coefficient[Series[((-1 + (1 - 25*x)^(-1/5))/5)^m, {x, 0, n}], x^n];
Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 38]]
(* Jean-François Alcover, Jun 21 2011, after g.f. *)

a(n, m) = 5*(5*(n-1)+m)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n

G.f. for m-th column: ((-1+(1-25*x)^(-1/5))/5)^m.

A132057 A convolution triangle of numbers obtained from A034904.

Original entry on oeis.org

1, 28, 1, 980, 56, 1, 37730, 2744, 84, 1, 1531838, 130340, 5292, 112, 1, 64337196, 6136956, 299782, 8624, 140, 1, 2766499428, 288408120, 16120314, 568008, 12740, 168, 1, 121034349975, 13561837212, 841627332, 34401528, 956970, 17640, 196, 1

Offset: 1

Wolfdieter Lang Sep 14 2007

a(n,1) = A034904(n). a(n,m)=: s2(8; n,m), a member of a sequence of unsigned triangles including s2(2; n,m)=A007318(n-1,m-1) (Pascal's triangle). s2(3;n,m)= A035324(n,m), s2(4; n,m)= A035529(n,m), s2(5; n,m)= A048882(n,m), s2(6; n,m)= A049375; s2(7; n,m)=A092083.

			{1}; {28,1}; {980,56,1}; (37730,2744,84,1);...

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First 10 rows.

Cf. A132058 (row sums), A132059 (negative of alternating row sums).

a[n_, m_] := a[n, m] = 7*(7*(n-1) + m)*a[n-1, m]/n + m*a[n-1, m-1]/n;
a[n_, m_] /; n < m = 0; a[_, 0] = 0; a[1, 1] = 1;
Flatten[Table[a[n, m], {n, 1, 8}, {m, 1, n}]][[1 ;; 36]]
(* Jean-François Alcover, Jun 17 2011 *)

a(n, m) = 7*(7*(n-1)+m)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n

G.f. for m-th column: ((-1+(1-49*x)^(-1/7))/7)^m.

A035330 5-fold convolution of A001700(n), n >= 0.

Original entry on oeis.org

1, 15, 140, 1045, 6835, 40963, 230720, 1240740, 6437890, 32468470, 160010280, 773624615, 3680728375, 17274086235, 80119845080, 367821324040, 1673528845710, 7554110698850, 33858536700040, 150802994850570

Offset: 0

Wolfdieter Lang

Fifth column of triangular array A035324.

Michael De Vlieger, Table of n, a(n) for n = 0..1650
José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, On One-Parameter Catalan Arrays, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

Cf. A000108, A045894, A035324.

Array[(#^2 + 27 # + 122) Binomial[2 (# + 5), # + 5]/24 - 5 (# + 8)*2^(2 # + 5) &, 20, 0] (* Michael De Vlieger, Sep 04 2018 *)

a(n) = (n^2+27*n+122)*binomial(2*(n+5), n+5)/24 - 5*(n+8)*2^(2*n+5) = A035324(n+5, 5);

G.f.: c(x)^5/(1-4*x)^(5/2), where c(x) = g.f. for Catalan numbers A000108.

A134283 A certain partition array in Abramowitz-Stegun (A-St)order, called M_0(3).

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 35, 20, 9, 9, 1, 126, 70, 60, 30, 27, 12, 1, 462, 252, 210, 100, 105, 180, 27, 40, 54, 15, 1, 1716, 924, 756, 700, 378, 630, 300, 270, 140, 360, 108, 50, 90, 18, 1, 6435, 3432, 2772, 2520, 1225, 1386, 2268, 2100, 945, 900, 504, 1260, 600, 1080, 81

Offset: 1

Wolfdieter Lang, Nov 13 2007

For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_0(3); the k=3 member in the family of a generalization of the multinomial number arrays M_0 = M_0(2) = A048996.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
The s2(3,n,m):=A035324(n,m) numbers (generalized Pascal triangle) are obtained by summing in row n all numbers with the same part number m. In the same manner the s2(2,n,m) = binomial(n-1,m-1) = A007318(n-1,m-1) numbers are obtained from the partition array M_0 = A048996.

			[1]; [3,1]; [10,6,1]; [35,20,9,9,1]; [126,70,60,30,27,12,1]; ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.

Cf. A049027 (row sums, also of triangle A035324).

a(n,k) = m!*Product_{j=1..n} (s2(3,j,1)^e(n,k,j))/e(n,k,j)! with s2(3,n,1) = A035324(n,1) = A001700(n-1) and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1.

A171488 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A005773(n+1)= 1,2,5,13,35,96,267,...

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 13, 14, 6, 1, 35, 46, 27, 8, 1, 96, 147, 107, 44, 10, 1, 267, 462, 396, 204, 65, 12, 1, 750, 1437, 1404, 858, 345, 90, 14, 1, 2123, 4438, 4835, 3388, 1625, 538, 119, 16, 1, 6046, 13637, 16305, 12802, 7072, 2805, 791, 152, 18, 1

Offset: 0

Philippe Deléham, Dec 10 2009

Equal to A064189*B = B*A054336 = B^(-1)*A035324, B = A007318.

			Triangle T(n,k) (0<=k<=n) begins:
   1;
   2,   1;
   5,   4,   1;
  13,  14,   6,  1;
  35,  46,  27,  8,  1;
  96, 147, 107, 44, 10, 1;
  ...

Cf. A097609, A064189.

T(n,k)=((k+1)*sum(binomial(2*j+k,j)*(-1)^j*3^(n-j-k)*binomial(n+1,j+k+1),j,0,n-k))/(n+1); /* Vladimir Kruchinin Sep 30 2020 */

Sum_{k, 0<=k<=n} T(n,k)*x^k = A005043(n), A001006(n), A005773(n+1), A059738(n) for x = -2, -1, 0, 1 respectively.
T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + sum_{i, i>=0} T(n-1,k+1+i)*(-1)^i. - Philippe Deléham, Feb 23 2012
T(n,k) = (k+1)*Sum_{j=0..n-k} C(2*j+k,j)*(-1)^j*3^(n-j-k)*C(n+1,j+k+1)/(n+1). - Vladimir Kruchinin Sep 30 2020

A134284 A certain partition array in Abramowitz-Stegun order (A-St order), called M_0(3)/M_0.

Original entry on oeis.org

1, 3, 1, 10, 3, 1, 35, 10, 9, 3, 1, 126, 35, 30, 10, 9, 3, 1, 462, 126, 105, 100, 35, 30, 27, 10, 9, 3, 1, 1716, 462, 378, 350, 126, 105, 100, 90, 35, 30, 27, 10, 9, 3, 1, 6435, 1716, 1386, 1260, 1225, 462, 378, 350, 315, 300, 126, 105, 100, 90, 81, 35, 30, 27, 10, 9, 3, 1

Offset: 1

Wolfdieter Lang, Nov 13 2007

The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_0(3)= A134283 with each entry divided by the corresponding one of the partition number array M_0 = M_0(2) = A048996; in short M_0(3)/M_0.

			[1]; [3,1]; [10,3,1]; [35,10,9,3,1]; [126,35,30,10,9,3,1]; ...
a(4,3) = 9 = 3^2 because (2^2) is the k=4 partition of n=4 in A-St order and s2(3,2,1)=3.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.

Cf. A134826 (row sums coinciding with those of triangle A134285).

a(n,k) = Product_{j=1..n} s2(3,j,1)^e(n,k,j) with s2(3,n,1) = A035324(n,1) = A001700(n-1) = binomial(2*n-1,n) and with the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

a(n,k) = A134283(n,k)/A048996(n,k) (division of partition arrays M_0(3) by M_0).

A154930 Inverse of Fibonacci convolution array A154929.

Original entry on oeis.org

1, -2, 1, 5, -4, 1, -15, 14, -6, 1, 51, -50, 27, -8, 1, -188, 187, -113, 44, -10, 1, 731, -730, 468, -212, 65, -12, 1, -2950, 2949, -1956, 970, -355, 90, -14, 1, 12235, -12234, 8291, -4356, 1785, -550, 119, -16, 1, -51822, 51821, -35643, 19474, -8612, 3021

Offset: 0

Paul Barry, Jan 17 2009

Alternating sign version of A104259. Row sums are (-1)^n*A033321. First column is (-1)^n*A007317.

			Triangle begins
1,
-2, 1,
5, -4, 1,
-15, 14, -6, 1,
51, -50, 27, -8, 1,
-188, 187, -113, 44, -10, 1,
731, -730, 468, -212, 65, -12, 1,
-2950, 2949, -1956, 970, -355, 90, -14, 1
Production array is
-2, 1,
1, -2, 1,
-1, 1, -2, 1,
1, -1, 1, -2, 1,
-1, 1, -1, 1, -2, 1,
1, -1, 1, -1, 1, -2, 1,
-1, 1, -1, 1, -1, 1, -2, 1
or ((1-x-x^2)/(1+x),x) beheaded.

Cf. A171567, A054336, A171488, A035324.

Riordan array ((1/(1+x))c(-x/(1+x)), (x/(1+x))c(x/(1+x))), c(x) the g.f. of A000108;
Riordan array ((sqrt(1+6x+5x^2)-x-1)/(2x(1+x)),(sqrt(1+6x+5x^2)-x-1)/ (2(1+x)));
Triangle T(n,k) = sum{j=0..n, (-1)^(n-k)*C(n,j)*C(2j-k,j-k)(k+1)/(j+1)}.
T(n,k) = T(n-1,k-1) -2*T(n-1,k) + Sum_{i, i>=0} T(n-1,k+1+i)*(-1)^i. - Philippe Deléham, Feb 23 2012

A116396 Expansion of 2/((2+x)sqrt(1-4x)-x).

Original entry on oeis.org

1, 2, 7, 25, 93, 353, 1358, 5273, 20614, 81003, 319584, 1264924, 5019743, 19963699, 79541181, 317406302, 1268283199, 5073605801, 20316709251, 81427911966, 326612013623, 1310968893954, 5265285993860, 21158914176719, 85071253608611

Offset: 0

Paul Barry, Feb 12 2006

Diagonal sums of number triangle A116395.

Diagonal sums of the Riordan matrix ((1-sqrt(1-4*x))/(2*x*sqrt(1-4*x)),(1-sqrt(1-4*x))/(2*sqrt(1-4*x))) (A035324). - Emanuele Munarini, Apr 26 2011

CoefficientList[Series[(x+(2+x)Sqrt[1-4x])/(2-6x-8x^2-2x^3),{x,0,25}],x] (* Emanuele Munarini, Apr 26 2011 *)

a(n) = Sum_{k=0..floor(n/2)} (4^(n-k)/2^k)*Sum_{j=0..k} C(k,j)*C(n-k+(j-1)/2,n-k)*(-1)^(k-j).

D-finite with recurrence: +2*n*a(n) +(-13*n+10)*a(n-1) +(9*n-16)*a(n-2) +2*(19*n-41)*a(n-3) +(23*n-66)*a(n-4) +2*(2*n-7)*a(n-5)=0. - R. J. Mathar, Jan 24 2020

A188110 Triangle T(n,m), [xA(x)]^m=sum(n>=m T(n,m)x^n), where A(x) satisfies xA(x)^3= -(2x*A(x)^2+sqrt(1-4xA(x)^2)-1)/(4xA(x)^2+sqrt(1-4xA(x)^2)-1).

Original entry on oeis.org

1, 3, 1, 28, 6, 1, 350, 65, 9, 1, 5020, 868, 111, 12, 1, 78023, 12924, 1581, 166, 15, 1, 1278340, 205766, 24468, 2516, 230, 18, 1, 21740636, 3428438, 399735, 40489, 3700, 303, 21, 1, 380161308, 59034600, 6784186, 679460, 61905, 5160, 385, 24, 1, 6792111260, 1042169972, 118444293, 11759612, 1067738, 89715, 6923, 476, 27, 1

Offset: 1

Vladimir Kruchinin, Mar 21 2011

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A035324.

(* S = A035324 *)
S[n_, m_] /; n >= m >= 1 := S[n, m] = 2(2(n-1) + m)(S[n - 1, m]/n) + m (S[n - 1, m - 1]/n); S[n_, m_] /; n < m = 0; S[n_, 0] = 0; S[1, 1] = 1;
T[n_, m_] := m/(2n-m) S[3n - 2m, 2n - m];
Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Feb 16 2019 *)

T(n,m) = m/(2*n-m)*A035324(3*n-2*m,2*n-m).

A188109 Triangle T(n,m), [xA(x)]^m=sum(n>=m T(n,m)x^n), where A(x) satisfies xA(x)^2= -(2x*A(x)+sqrt(1-4xA(x))-1)/(4xA(x)+sqrt(1-4xA(x))-1).

Original entry on oeis.org

1, 3, 1, 19, 6, 1, 152, 47, 9, 1, 1367, 418, 84, 12, 1, 13195, 4007, 825, 130, 15, 1, 133556, 40368, 8433, 1400, 185, 18, 1, 1398696, 421332, 88872, 15239, 2170, 249, 21, 1, 15029311, 4515706, 959080, 168112, 25100, 3162, 322, 24, 1, 164764985, 49405895, 10547361, 1878462, 289788, 38772, 4403, 404, 27, 1

Offset: 1

Vladimir Kruchinin, Mar 21 2011

			1;
3, 1;
19, 6, 1;
152, 47, 9, 1;
1367, 418, 84, 12, 1;
13195, 4007, 825, 130, 15, 1;
133556, 40368, 8433, 1400, 185, 18, 1;
1398696, 421332, 88872, 15239, 2170, 249, 21, 1

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A035324

(* S = A035324 *)
S[n_, m_] /; n >= m >= 1 := S[n, m] = 2(2(n-1)+m)(S[n-1, m]/n) + m(S[n-1, m-1]/n); S[n_, m_] /; n < m = 0; S[n_, 0] = 0; S[1, 1] = 1;
T[n_, m_] := m/n S[2n-m, n];
Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Feb 16 2019 *)

T(n,m) = m/n*A035324(2*n-m,n).

Previous Showing 11-20 of 20 results.

cp's OEIS Frontend

A049375 A convolution triangle of numbers obtained from A034687.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A132057 A convolution triangle of numbers obtained from A034904.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A035330 5-fold convolution of A001700(n), n >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A134283 A certain partition array in Abramowitz-Stegun (A-St)order, called M_0(3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A171488 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A005773(n+1)= 1,2,5,13,35,96,267,...

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maxima

Formula

A134284 A certain partition array in Abramowitz-Stegun order (A-St order), called M_0(3)/M_0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A154930 Inverse of Fibonacci convolution array A154929.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A116396 Expansion of 2/((2+x)*sqrt(1-4*x)-x).

Views

Author

Keywords

Comments

Programs

Mathematica

Formula

A188110 Triangle T(n,m), [x*A(x)]^m=sum(n>=m T(n,m)*x^n), where A(x) satisfies x*A(x)^3= -(2*x*A(x)^2+sqrt(1-4*x*A(x)^2)-1)/(4*x*A(x)^2+sqrt(1-4*x*A(x)^2)-1).

Views

Author

A116396 Expansion of 2/((2+x)sqrt(1-4x)-x).

A188110 Triangle T(n,m), [xA(x)]^m=sum(n>=m T(n,m)x^n), where A(x) satisfies xA(x)^3= -(2x*A(x)^2+sqrt(1-4xA(x)^2)-1)/(4xA(x)^2+sqrt(1-4xA(x)^2)-1).

A188109 Triangle T(n,m), [xA(x)]^m=sum(n>=m T(n,m)x^n), where A(x) satisfies xA(x)^2= -(2x*A(x)+sqrt(1-4xA(x))-1)/(4xA(x)+sqrt(1-4xA(x))-1).