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| A140882
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Triangle by rows with row n formed by coefficients of the characteristic polynomial of the n X n tridiagonal matrix with m_{i,i} = 2 for i=1..n, m_{i,i-1} = m_{i,i+1} = -1 for i=2..n-1, and m_{1,2} = m_{n,n-1} = -2.
(history;
published version)
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#22 by Tom Copeland at Fri Apr 26 19:38:17 EDT 2024
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#21 by Tom Copeland at Fri Apr 26 19:33:40 EDT 2024
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| FORMULA
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A generating function for the third and later shifted, unsigned rows is (4 + t) / (1 - (2 + t) x + x^2) = Sum_ (n >= 0) (4+t) U_n((2 + t)/2) x^n = 4 + t + (8 + 6t + t^2) x + ..., where U_n(t) are the Chebyshev polynomials of the second kind of A133156. U_n((2 + t)/2) = V_n((2 + t)), where V_n(t) are the Chebyshev polynomials of A049310. A formula for the coefficients is given by Eqn. 8 in Haukkanen et al. - Tom Copeland, Apr 26 2024
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Discussion
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| Fri Apr 26
| 19:38
| Tom Copeland: A g.f. was requested by Prof. Karol Penson, so I include it here.
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#20 by Tom Copeland at Fri Apr 26 19:25:14 EDT 2024
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| FORMULA
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A generating function for the third and later shifted, unsigned rows is (4 + t) / (1 - (2 + t) x + x^2) = Sum_ (n >= 0) (4+t) U_n((2 + t)/2) x^n = 4 + t + (8 + 6t + t^2) x + ..., where U_n(t) are the Chebyshev polynomials of the second kind of A133156. U_n((2 + t)/2) = V_n((2 + t)), where V_n(t) are the Chebyshev polynomials of A049310. - Tom Copeland, Apr 26 2024
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| CROSSREFS
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Cf. A049310, A133156.
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| STATUS
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approved
editing
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#46 by Tom Copeland at Mon Apr 01 15:13:34 EDT 2024
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Discussion
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| Mon Apr 01
| 16:01
| Peter Luschny: There is a lot of verbiage and too little care to read even two lines of history correctly. Your first change: "Closed parenthesis typo removed by Andrey Zabolotskiy, Mar 07 2024" was incorrect. The superfluous parenthesis was removed on 20 Mar. You ignored the change on Mar 07; after Joerg's hint, now the turnaround. Why don't you take some time out? It'll do you good, and stop wasting our time with this kind of nitpicking.
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| 16:03
| Andrew Howroyd: Note that the Extensions field is used to record changes to the Data, Offset and Name fields only and sometimes also other major edits (say affecting multiple comments and formulas). Typos and corrections to single formulas or comments are not typically noted in the Extensions field if at all. [Editors in particular often make small changes without any attribution, but material changes to a say a formula should have the change recorded next to the formula rather than in the Extensions field]. In this case there was a change to the Data field, so that is what is being recorded. (If a single term is being corrected then we might write a(4) corrected, but here there is more than one - anyone wanting the details needs to examine the history - this is just a note that the Data field was modified on the specified date). The main use case is when discrepancies are discovered between related sequences - sometimes a correction is made to 1 sequence, but it is forgotten to correct another until inconsistencies are discovered later. I hope that makes the purpose of the Extensions field clearer. [Above all it is not an alternative to history, which is the only accurate record of changes].
I'll review this - it is fine (too unimportant to discuss further).
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#45 by Tom Copeland at Mon Apr 01 15:09:47 EDT 2024
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| EXTENSIONS
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CorrectedData corrected by Andrey Zabolotskiy, Mar 07 2024
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| STATUS
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proposed
editing
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Discussion
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| Mon Apr 01
| 15:12
| Tom Copeland: Actually takes more effort to see how the data was changed, but this can be re-checked by anyone with a symbolic math app. In any event, I thank AZ for the effort in catching the errors.
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#42 by Tom Copeland at Mon Apr 01 09:36:51 EDT 2024
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Discussion
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| Mon Apr 01
| 10:22
| Joerg Arndt: Nope, data was wrong as well, see https://oeis.org/history?seq=A267633&start=10
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#41 by Tom Copeland at Mon Apr 01 09:35:56 EDT 2024
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| EXTENSIONS
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CorrectedClosed parenthesis typo removed by Andrey Zabolotskiy, Mar 07 2024
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| STATUS
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approved
editing
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| A356146
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Coefficients of the partition polynomials that are binomial convolutions of the partition polynomials of A133314, the refined Euler characteristic polynomials of the permutahedra and coefficient polynomials of reciprocals of Taylor series or e.g.f.s. Irregular triangle read by rows with length given by A000041.
(history;
published version)
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#16 by Tom Copeland at Sat Mar 23 15:26:18 EDT 2024
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#15 by Tom Copeland at Sat Mar 23 15:23:04 EDT 2024
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| EXTENSIONS
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Ordering in row 7 and formulag.f. changed correctedto e.g.f. by Andrey Zabolotskiy, Mar 07 2024
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| STATUS
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approved
editing
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Discussion
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| Sat Mar 23
| 15:25
| Tom Copeland: Andrey, not a correction, but a better, more easily understood presentation as an e.g.f
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| A239473
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Triangle read by rows: signed version of A059260: coefficients for expansion of partial sums of sequences a(n,x) in terms of their binomial transforms (1+a(.,x))^n ; Laguerre polynomial expansion of the truncated exponential.
(history;
published version)
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#129 by Tom Copeland at Fri Mar 08 11:58:24 EST 2024
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Discussion
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| Fri Mar 08
| 12:45
| Thomas Scheuerle: Nice to see you here again.
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