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Search: a134094 -id:a134094
Displaying 1-7 of 7 results found. page 1
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A211210 a(n) = Sum_{k=0..n} binomial(n, k)*|S1(n, k)|. +10
8
1, 1, 3, 16, 115, 1021, 10696, 128472, 1734447, 25937683, 424852351, 7554471156, 144767131444, 2971727661124, 65013102375404, 1509186299410896, 37032678328740751, 957376811266995031, 25999194631060525009, 739741591417352081464, 22000132609456951524051 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Binomial convolution of the unsigned Stirling numbers of the first kind.
Row sums of triangle A187555.
LINKS
Table of n, a(n) for n=0..20.
MATHEMATICA
Table[Sum[Binomial[n, k] Abs[StirlingS1[n, k]], {k, 0, n}], {n, 0, 20}]
PROG
(PARI) a(n) = sum(k=0, n, binomial(n, k)*abs(stirling(n, k, 1))); \\ Michel Marcus, May 10 2021
CROSSREFS
Cf. A317274 (signed S1), A187555, A134090, A211211.
Cf. A122455 (second kind), A271702, A134094, A343841 (second kind inverse).
KEYWORD
nonn
AUTHOR
Olivier Gérard, Oct 23 2012
STATUS
approved
A134090 Triangle, read by rows, where T(n,k) = [(I + D*C)^n](n,k); that is, row n of T = row n of (I + D*C)^n for n>=0 where C denotes Pascal's triangle, I the identity matrix and D a matrix where D(n+1,n)=1 and zeros elsewhere. +10
6
1, 1, 1, 3, 2, 1, 13, 9, 3, 1, 71, 46, 18, 4, 1, 456, 285, 110, 30, 5, 1, 3337, 2021, 780, 215, 45, 6, 1, 27203, 16023, 6167, 1729, 371, 63, 7, 1, 243203, 139812, 53494, 15176, 3346, 588, 84, 8, 1, 2357356, 1326111, 504030, 143814, 32376, 5886, 876, 108, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Column 0 equals A122455 if we define A122455(0)=1.
LINKS
Table of n, a(n) for n=0..54.
FORMULA
T(n,k) = [x^(n-k)] Sum_{j=0..n} C(n,j)*x^j/(1-j*x)^k /[Product_{i=0..j}(1-i*x)].
EXAMPLE
Triangle T begins:
1;
1, 1;
3, 2, 1;
13, 9, 3, 1;
71, 46, 18, 4, 1;
456, 285, 110, 30, 5, 1;
3337, 2021, 780, 215, 45, 6, 1;
27203, 16023, 6167, 1729, 371, 63, 7, 1;
243203, 139812, 53494, 15176, 3346, 588, 84, 8, 1;
2357356, 1326111, 504030, 143814, 32376, 5886, 876, 108, 9, 1; ...
Let P denote the matrix equal to Pascal's triangle shift down 1 row:
P(n,k) = C(n+1,k) for n>k>=0, with P(n,n)=1 for n>=0.
Illustrate row n of T = row n of P^n as follows.
Matrix P = I + D*C begins:
1;
1, 1;
1, 1, 1;
1, 2, 1, 1;
1, 3, 3, 1, 1;
1, 4, 6, 4, 1, 1; ...
Matrix cube P^3 begins:
1;
3, 1;
6, 3, 1;
13, 9, 3, 1; <== row 3 of P^3 = row 3 of T
30, 25, 12, 3, 1;
73, 72, 40, 15, 3, 1; ...
Matrix 4th power P^4 begins:
1;
4, 1;
10, 4, 1;
26, 14, 4, 1;
71, 46, 18, 4, 1; <== row 4 of P^4 = row 4 of T
204, 155, 70, 22, 4, 1; ...
Matrix 5th power P^5 begins:
1;
5, 1;
15, 5, 1;
45, 20, 5, 1;
140, 75, 25, 5, 1;
456, 285, 110, 30, 5, 1; <== row 5 of P^5 = row 5 of T.
PROG
(PARI) /* As generated by the g.f.: */ {T(n, k)=polcoeff(sum(j=0, n, binomial(n, j)*x^j/(1-j*x)^k/prod(i=0, j, 1-i*x+x*O(x^(n-k)))), n-k)} /* As generated by matrix power: row n of T equals row n of P^n: */ {T(n, k)=local(P=matrix(n+1, n+1, r, c, if(r==c, 1, if(r>c, binomial(r-2, c-1))))); (P^n)[n+1, k+1]}
CROSSREFS
Cf. columns: A134091, A134092, A134093; A134094 (row sums).
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Oct 07 2007
STATUS
approved
A134091 Column 1 of triangle A134090. +10
4
1, 2, 9, 46, 285, 2021, 16023, 139812, 1326111, 13544857, 147880458, 1715413558, 21036674321, 271585117428, 3677831536291, 52081368845176, 769123715337395, 11816582501728389, 188470925178659344, 3114771205613655362 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Row n of triangle T=A134090 = row n of (I + D*C)^n for n>=0 where C denotes Pascal's triangle, I the identity matrix and D a matrix where D(n+1,n)=1 and zeros elsewhere.
LINKS
Table of n, a(n) for n=0..19.
FORMULA
a(n) = [x^n] Sum_{k=0..n+1} C(n+1,k)*x^k/(1-k*x) / [Product_{i=1..k}(1-i*x)].
PROG
(PARI) a(n)=polcoeff(sum(k=0, n+1, binomial(n+1, k)*x^k/(1-k*x)/prod(i=0, k, 1-i*x +x*O(x^n))), n)
CROSSREFS
Cf. A134090; columns: A122455, A134092, A134093; A134094 (row sums); A048993 (S2).
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 07 2007
STATUS
approved
A134092 Column 2 of triangle A134090. +10
4
1, 3, 18, 110, 780, 6167, 53494, 504030, 5112090, 55411697, 638154165, 7770348170, 99618149267, 1339889000543, 18848892749144, 276573551651632, 4222814264496510, 66947348027905977, 1099955438013660173 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Row n of triangle T=A134090 = row n of (I + D*C)^n for n>=0 where C denotes Pascal's triangle, I the identity matrix and D a matrix where D(n+1,n)=1 and zeros elsewhere.
LINKS
Table of n, a(n) for n=0..18.
FORMULA
a(n) = [x^n] Sum_{k=0..n+2} C(n+2,k)*x^k/(1-k*x)^2 / [Product_{i=1..k}(1-i*x)].
PROG
(PARI) {a(n)= polcoeff(sum(k=0, n+2, binomial(n+2, k)*x^k/(1-k*x)^2/prod(i=0, k, 1-i*x +x*O(x^n))), n)}
CROSSREFS
Cf. A134090; columns: A122455, A134091, A134093; A134094 (row sums); A048993 (S2).
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 08 2007
STATUS
approved
A134093 Column 3 of triangle A134090. +10
4
1, 4, 30, 215, 1729, 15176, 143814, 1462995, 15876410, 182811992, 2223580281, 28458251185, 381943459065, 5359649816728, 78430018675440, 1194057733357517, 18873870914263424, 309154787519651284, 5238840625331179517 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Row n of triangle T=A134090 = row n of (I + D*C)^n for n>=0 where C denotes Pascal's triangle, I the identity matrix and D a matrix where D(n+1,n)=1 and zeros elsewhere.
LINKS
Table of n, a(n) for n=0..18.
FORMULA
a(n) = [x^n] Sum_{k=0..n+3} C(n+3,k)*x^k/(1-k*x)^3 / [Product_{i=1..k}(1-i*x)].
PROG
(PARI) {a(n)= polcoeff(sum(k=0, n+3, binomial(n+3, k)*x^k/(1-k*x)^3/prod(i=0, k, 1-i*x +x*O(x^n))), n)}
CROSSREFS
Cf. A134090; columns: A122455, A134091, A134092; A134094 (row sums); A048993 (S2).
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 08 2007
STATUS
approved
A211211 sum( C(n+1,k)*|S1(n,k)|, k=0..n). Binomial convolution of the Stirling numbers of the first kind. +10
3
1, 2, 6, 30, 205, 1750, 17766, 207942, 2746815, 40315858, 649688072, 11387466948, 215440517656, 4371810051908, 94649397546302, 2176321870192342, 52938365091640943, 1357592080006964806, 36593629200726397630, 1033979281229140895582, 30552322294916306960625 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..20.
MATHEMATICA
Table[Sum[Binomial[n + 1, k] Abs[StirlingS1[n, k]], {k, 0, n}], {n, 0, 20}]
CROSSREFS
Shifted version of A211210. S1 Analog of A134094.
KEYWORD
nonn
AUTHOR
Olivier Gérard, Oct 23 2012
STATUS
approved
A271702 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j-1,-n-1)*S2(k,j), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n. +10
1
1, 1, 1, 1, 2, 3, 1, 3, 6, 13, 1, 4, 10, 26, 71, 1, 5, 15, 45, 140, 456, 1, 6, 21, 71, 246, 887, 3337, 1, 7, 28, 105, 399, 1568, 6405, 27203, 1, 8, 36, 148, 610, 2584, 11334, 51564, 243203, 1, 9, 45, 201, 891, 4035, 18849, 91101, 455712, 2357356 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Table of n, a(n) for n=0..54.
FORMULA
T(n,k) = Sum_{j=0..k} C(n,j) * S2(k,j). - Alois P. Heinz, Sep 03 2019
EXAMPLE
Triangle starts:
[1]
[1, 1]
[1, 2, 3]
[1, 3, 6, 13]
[1, 4, 10, 26, 71]
[1, 5, 15, 45, 140, 456]
[1, 6, 21, 71, 246, 887, 3337]
[1, 7, 28, 105, 399, 1568, 6405, 27203]
MAPLE
T := (n, k) -> add(Stirling2(k, j)*binomial(-j-1, -n-1)*(-1)^(n-j), j=0..n):
seq(seq(T(n, k), k=0..n), n=0..9);
MATHEMATICA
Flatten[Table[Sum[(-1)^(n-j) Binomial[-j-1, -n-1] StirlingS2[k, j], {j, 0, n}], {n, 0, 9}, {k, 0, n}]]
CROSSREFS
A000012 (col. 0), A000027 (col. 1), A000217 (col. 2), A008778 (col. 3), A122455 (diag. n,n), A134094 (diag. n,n-1).
Cf. A048993.
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Apr 14 2016
STATUS
approved
page 1

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Last modified August 29 03:06 EDT 2024. Contains 375510 sequences. (Running on oeis4.)