The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A248600 G.f.: Sum_{n>=0} R_n(x+x*y) * x^(2*n)*y^n / (1-x-x*y)^(4*n+1) = Sum_{n>=0} Sum_{k=0..n} C(n,k)^4 * x^n*y^k, where R_n(x+x*y) equals the n-th row polynomial R_n(z) = Sum_{k=0..2*n} T(n,k)*z^k at z = x+x*y. 1
 1, 14, 8, 2, 786, 1056, 576, 96, 6, 61340, 131760, 117900, 48320, 9540, 720, 20, 5562130, 16481920, 20917120, 13847680, 5118400, 1025920, 105280, 4480, 70, 549676764, 2079579600, 3444581700, 3165926400, 1755532800, 598123008, 123656400, 14716800, 926100, 25200, 252, 57440496036 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Table of n, a(n) for n=0..36. FORMULA Leftmost border equals A050983, de Bruijn's S(4,n): T(n,0) = Sum_{k=0..2*n} (-1)^(n+k) * C(2*n,k)^4. Rightmost border equals A000984, the central binomial coefficients: T(n,2*n) = Sum_{k=0..2*n} (-1)^(n+k)* C(2*n,k)^2 = (2*n)!/(n!)^2. Row sums equal A008977(n) = (4*n)!/(n!)^4. Sum_{k=0..n} (-1)^k * T(n,k) = A002897(n) = C(2*n,n)^3. EXAMPLE Triangle begins: [1], [14, 8, 2], [786, 1056, 576, 96, 6], [61340, 131760, 117900, 48320, 9540, 720, 20], [5562130, 16481920, 20917120, 13847680, 5118400, 1025920, 105280, 4480, 70], [549676764, 2079579600, 3444581700, 3165926400, 1755532800, 598123008, 123656400, 14716800, 926100, 25200, 252], [57440496036, 264565490112, 542687590368, 640299696960, 477284304420, 233110386432, 75243589344, 15835792896, 2103157980, 165802560, 7051968, 133056, 924], [6242164112184, 33895475918304, 83073660613944, 119912994225024, 112698387745944, 72172565713248, 32111980788888, 9951304416768, 2124873478728, 305035899168, 28270554312, 1584815232, 48600552, 672672, 3432], [698300344311570, 4368053451041280, 12465205610457600, 21305587665922560, 24216302627637120, 19255941998092800, 10989839486545920, 4550117424652800, 1366687981264320, 295074717949440, 44954858108160, 4691645038080, 320878958400, 13445752320, 311351040, 3294720, 12870], ... where this triangle forms the coefficients in the series B(x,y) = 1/(1-x-x*y) + (14 + 8*(x+x*y) + 2*(x+x*y)^2) * x^2*y/(1-x-x*y)^5 + (786 + 1056*(x+x*y) + 576*(x+x*y)^2 + 96*(x+x*y)^3 + 6*(x+x*y)^4) * x^4*y^2/(1-x-x*y)^9 + (61340 + 131760*(x+x*y) + 117900*(x+x*y)^2 + 48320*(x+x*y)^3 + 9540*(x+x*y)^4 + 720*(x+x*y)^5 + 20*(x+x*y)^6) * x^6*y^3/(1-x-x*y)^13 +... such that the sum may be expressed using binomial coefficients C(n,k)^4 like so: B(x,y) = 1 + x*(1 + y) + x^2*(1 + 2^4*y + y^2) + x^3*(1 + 3^4*y + 3^4*y^2 + y^3) + x^4*(1 + 4^4*y + 6^4*y^2 + 4^4*y^3 + y^4) + x^5*(1 + 5^4*y + 10^4*y^2 + 10^4*y^3 + 5^4*y^4 + y^5) + x^6*(1 + 6^4*y + 15^4*y^2 + 20^4*y^3 + 15^4*y^4 + 6^4*y^5 + y^6) +... The central terms of the rows begin: [1, 8, 576, 48320, 5118400, 598123008, 75243589344, 9951304416768, 1366687981264320, ...]. CROSSREFS Cf. A050983, A000984, A008977, A002897, A187056, A248706. Sequence in context: A275337 A337420 A154037 * A240245 A119871 A245175 Adjacent sequences: A248597 A248598 A248599 * A248601 A248602 A248603 KEYWORD nonn,tabf AUTHOR Paul D. Hanna, Oct 11 2014 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 18 03:29 EDT 2024. Contains 373468 sequences. (Running on oeis4.)