A000346 -id:A000346 - cp's OEIS frontend

A375086 Row sums of A375085.

0, 1, 3, 8, 24, 78, 268, 956, 3496, 12998, 48876, 185268, 706456, 2706204, 10404696, 40124792, 155133904, 601113158, 2333671756, 9075266372, 35345525944, 137847053108, 538258923016, 2104101060872, 8233434921904, 32247612071708, 126410623214968, 495918566502536

Offset: 0

Stefano Spezia, Jul 29 2024

nonn

Cf. A375085.

Cf. A000079, A000302, A000346, A000984, A001791, A007582.

a[n_]:=(2^(n+1)+4^n+4Binomial[2n-2,n-1]-8Binomial[2n-2,n-2]Hypergeometric2F1[1,2-n,1+n,-1])/8; Join[{0,1},Array[a,26,2]]

a(n) = (2^(n+1) + 4^n + 4*binomial(2*n-2,n-1) - 8*binomial(2*n-2,n-2)*hypergeom([1, 2-n], [1+n], -1))/8 for n > 1.

a(n) ~ A007582(n+2).

A090299 Table T(n,k), n>=0 and k>=0, read by antidiagonals : the k-th column given by the k-th polynomial K_k related to A090285.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 10, 5, 1, 14, 35, 22, 7, 1, 42, 126, 93, 38, 9, 1, 132, 462, 386, 187, 58, 11, 1, 429, 1716, 1586, 874, 325, 82, 13, 1, 1430, 6435, 6476, 3958, 1686, 515, 110, 15, 1, 4862, 24310, 26333, 17548, 8330, 2934, 765, 142, 17, 1

Offset: 0

Philippe Deléham, Jan 25 2004

Read as a number triangle, this is the Riordan array (c(x),x/sqrt(1-4x)) where c(x) is the g.f. of A000108. - Paul Barry, May 16 2005

			row n=0 : 1, 1, 2, 5, 14, 42, 132, 429, ... see A000108.
row n=1 : 1, 3, 10, 35, 126, 462, 1716, 6435, ... see A001700.
row n=2 : 1, 5, 22, 93, 386, 1586, 6476, ... see A000346.
row n=3 : 1, 7, 38, 187, 874, 3958, 17548, ... see A000531.
row n=4 : 1, 9, 58, 325, 1686, 8330, 39796, ... see A018218.

Other rows : A029887, A042941, A045724, A042985, A045492. Columns : A000012, A005408. Row n is the convolution of the row (n-j) with A000984, A000302, A002457, A002697 (first term omitted), A002802, A038845, A020918, A038846, A020920 for j=1, 2, ..9 respectively.

T(n, k) = K_k(n)= Sum_{j>=0} A090285(k, j)*2^j*binomial(n, j). T(n, 1) = 2*n+1. T(n, 2) = 2*A028387(n).

Corrected by Alford Arnold, Oct 18 2006

A204451 2*A014335 - A203578. Difference of the exponential convolution of A000045 (Fibonacci) with itself and the corresponding exponential half-convolution.

Original entry on oeis.org

0, 0, 0, 3, 8, 35, 75, 371, 888, 3891, 9445, 40755, 102323, 426803, 1091167, 4469555, 11625960, 46805811, 123364443, 490156851, 1306737465, 5132989235, 13816838695, 53753361203, 145912841523, 562912506675, 1539304050375, 5894896300851, 16225419029303, 61732155503411, 170909837010835

Offset: 0

Wolfdieter Lang, Jan 16 2012

nonn

See A203578 for the exponential (or binomial) half-convolution of A000045 (Fibonacci). The present sequence has to be added to this sequence in order to obtain the (full) exponential convolution 2*A014335.

Cf. A000045, A014335, A203578.

a(n) = sum(binomial(n,k)*F(k)*F(n-k), k=floor(n/2)+1..n), n>=0, with the Fibonacci numbers A000045(n).
E.g.f.: exp(x)*(cosh((2*phi-1)*x)-1)/5 - (BesselI(0,2*phi*x) + BesselI(0,2*(phi-1)*x) - 2*BesselI(0,2*I*x))/10. See the e.g.f. of A203578, also for phi and BesselI.
Bisection: a(2*k) =((2^(2*k) - binomial(2*k,k))*L(2*k)/2 - (1 - (-1)^k*binomial(2*k,k)))/5 and a(2*k+1) = (2^(2*k)*L(2*k+1) - 1)/5 = A203578(2*k), k>=0, with the Lucas numbers L(n)= A000032(n). Compare with A203578. See A000346(k-1), with A000346(-1)=0, for (2^(2*k) - binomial(2*k,k))/2, k>=0.

A355635 Triangle read by rows. Row n gives the coefficients of Product_{k=0..n-1} (x - binomial(n-1,k)) expanded in decreasing powers of x, with row 0 = {1}.

Original entry on oeis.org

1, 1, -1, 1, -2, 1, 1, -4, 5, -2, 1, -8, 22, -24, 9, 1, -16, 93, -238, 256, -96, 1, -32, 386, -2180, 5825, -6500, 2500, 1, -64, 1586, -19184, 117561, -345600, 407700, -162000, 1, -128, 6476, -164864, 2229206, -15585920, 51583084, -64538880, 26471025

Offset: 0

Thomas Scheuerle, Jul 11 2022

Without signs the triangle of elementary symmetric functions of the terms binomial(n,j), j=0..n.

			The triangle begins:
  1;
  1,  -1;
  1,  -2,   1;
  1,  -4,   5,    -2;
  1,  -8,  22,   -24,    9;
  1, -16,  93,  -238,  256,   -96;
  1, -32, 386, -2180, 5825, -6500, 2500;
  ...
Row 4: x^4 - 8*x^3 + 22*x^2 - 24*x + 9 = (x-1)*(x-4)*(x-6)*(x-4)*(x-1).

Cf. A000079, A000346, A025131, A025133, A025134, A025135.

Cf. A001142 (right diagonal unsigned).

T(n, k) = polcoeff(prod(m=0, n, (x-binomial(n-1, m))), n-k+1);

T(n, 0) = 1.
T(n, 1) = -2^(n-1), for n > 0.
T(n, 2) = A000346(n-2), for n > 1.
T(n, 3) = -A025131(n-1), for n > 1.
T(n, 4) = A025133(n-1), for n > 1.
T(n, n) = (-1)^n*A001142(n-1), for n > 0.
T(n+1, n) = (-1)^n*A025134(n).
T(n+2, n) = (-1)^n*A025135(n).

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A375086 Row sums of A375085.

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A090299 Table T(n,k), n>=0 and k>=0, read by antidiagonals : the k-th column given by the k-th polynomial K_k related to A090285.

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A204451 2*A014335 - A203578. Difference of the exponential convolution of A000045 (Fibonacci) with itself and the corresponding exponential half-convolution.

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Author

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Comments

Crossrefs

Formula

A355635 Triangle read by rows. Row n gives the coefficients of Product_{k=0..n-1} (x - binomial(n-1,k)) expanded in decreasing powers of x, with row 0 = {1}.

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