{"id":"n-177268549ede7415","revision":0,"first_index":0,"fingerprint":"f6-4ffec972130b4198","name":"T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n), triangle read by rows, n >= 0 and 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["0","1"],["0","1","12"],["0","1","60","360"],["0","1","252","5040","20160"],["0","1","1020","52920","604800","1814400"],["0","1","4092","506880","12640320","99792000","239500800"],["0","1","16380","4684680","230630400","3632428800","21794572800","43589145600"],["0","1","65532","42653520","3952428480","111567456000","1264085222400","6102480384000","10461394944000"],["0","1","262140","385847640","65627452800","3144904963200","60233972198400","529085049292800","2134124568576000","3201186852864000"],["0","1","1048572","3480492960","1071646712640","84529094832000","2583550454284800","36887750355456000","263564384219136000","912338253066240000","1216451004088320000"],["0","1","4194300","31355893800","17341255008000","2209675574937600","104165656872076800","2277705950097177600","25721180675334144000","154550100069421056000","468333636574003200000","562000363888803840000"],["0","1","16777212","282328873200","279216010180800","56802602324160000","4041640823286528000","130565741105479680000","2192804991244707840000","20389239392275353600000","105562401683780321280000","284372184127734743040000","310224200866619719680000"],["0","1","67108860","2541463175160","4483266579792000","1445194499020272000","152997013145104128000","7133299944006652416000","171675297304976424960000","2322526718095184240640000","18304151137442666496000000","83178863857362412339200000","201645730563302817792000000","201645730563302817792000000"],["0","1","268435452","22875181842240","71874587214480960","36533356467688080000","5698658147094424512000","377377153648734919680000","12699211014080087777280000","240657305141032709529600000","2712975266620436661043200000","18521160352239363814195200000","74951718050379657373286400000","165147853331345007771648000000","152444172305856930250752000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","13","421","25453","2473141","352444093","69251478661","17943523153933","5927841361456981","2431910546406522973","1212989379862721528101","722875495525684291639213","507275965883448333971692021","414031618935013558427928710653"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A304330","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-2"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-10"],["0","1","10"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","-4"],["0","0","1","4"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":170036,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-13588e30a33fe9a4"]},{"id":170037,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-13588e30a33fe9a4"]},{"id":170038,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-13588e30a33fe9a4"]}],"region_exact":{},"incoming_edges":[{"id":"e-13588e30a33fe9a4","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-177268549ede7415","premise_count":0,"theorem":"Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; S_n = (A + B D + C D^2 with A = -2 + 2 n, B = -1 + n - 10 x + 10 n x, C = -x + n x - 4 x^2 + 4 n x^2) / (n - 1): Branden's Weyl step: T = A + B D + C D^2 maps every real-rooted polynomial to a real-rooted polynomial when (i) B^2 - 4 A C >= 0 on the real line, (ii) A strictly interlaces B (deg B = deg A + 1) or strictly alternates left of B (deg B = deg A), with leading coefficients of the same sign when deg A >= 1, and (iii) T(x^k) has degree deg A + k with leading coefficients of one sign for every k >= 0 (P. Branden, Linear transformations preserving the Polya frequency property, Trans. Amer. Math. Soc. 358 (2006), Corollary 3.5); (i) is necessary for a real-rootedness preserver (J. Borcea and P. Branden, Proc. London Math. Soc. 101 (2010), Theorem 1.3); with coefficients of one sign T also keeps every root in (-inf, 0]","lean_route":null,"lean_declarations":[],"certificate_hash":"4d02c35982461f6c370c2984de7d0bb401c1b07c49875312c761de7ea0c83b87","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]}],"used_by":[{"id":"e-32f63ae67c0775fd","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-9239b1466464de00","conclusion_name":"derivative of T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n), triangle read by rows, n >= 0 and 0 <= k <= n."}],"used_by_total":1,"links":[{"id":"l-833f797117ead319","kind":"oeis_match","node_a_id":"n-177268549ede7415","node_b_id":null,"fingerprint":null,"oeis_id":"A304330","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A304330","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-abb152b527d5043c","kind":"oeis_match","node_a_id":"n-177268549ede7415","node_b_id":null,"fingerprint":null,"oeis_id":"A100872","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A100872","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A304330","relation":"identification"},{"oeis_id":"A100872","relation":"match_row_sums"},{"oeis_id":"A304330","relation":"match_triangle"}],"bfiles":[{"node_id":"n-177268549ede7415","oeis_id":"A100872","target":"row_sums","oeis_offset":1,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":15,"oeis_terms_available":15,"term_count":200,"raw_size":80354},{"node_id":"n-177268549ede7415","oeis_id":"A304330","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":64442}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 0, 1;\n  2: 0, 1,     12;\n  3: 0, 1,     60,       360;\n  4: 0, 1,    252,      5040,       20160;\n  5: 0, 1,   1020,     52920,      604800,       1814400;\n  6: 0, 1,   4092,    506880,    12640320,      99792000,      239500800;\n  7: 0, 1,  16380,   4684680,   230630400,    3632428800,    21794572800,     43589145600;\n  8: 0, 1,  65532,  42653520,  3952428480,  111567456000,  1264085222400,   6102480384000,   10461394944000;\n  9: 0, 1, 262140, 385847640, 65627452800, 3144904963200, 60233972198400, 529085049292800, 2134124568576000, 3201186852864000;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2)*x*P(n-1,x) + ((n-1)*x+(10*n-10)*x^2)*d/dx P(n-1,x) + ((n-1)*x^2+(4*n-4)*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2)*x*P[n - 1, x] + ((n - 1)*x + (10*n - 10)*x^2)*D[P[n - 1, x], x] + ((n - 1)*x^2 + (4*n - 4)*x^3)*D[P[n - 1, x], {x, 2}])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2)*x*v[i - 1] + ((n - 1)*x + (10*n - 10)*x^2)*deriv(v[i - 1], x) + ((n - 1)*x^2 + (4*n - 4)*x^3)*deriv(deriv(v[i - 1], x), x))/(n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A304330","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 0, 1;\n  2: 0, 1,     12;\n  3: 0, 1,     60,       360;\n  4: 0, 1,    252,      5040,       20160;\n  5: 0, 1,   1020,     52920,      604800,       1814400;\n  6: 0, 1,   4092,    506880,    12640320,      99792000,      239500800;\n  7: 0, 1,  16380,   4684680,   230630400,    3632428800,    21794572800,     43589145600;\n  8: 0, 1,  65532,  42653520,  3952428480,  111567456000,  1264085222400,   6102480384000,   10461394944000;\n  9: 0, 1, 262140, 385847640, 65627452800, 3144904963200, 60233972198400, 529085049292800, 2134124568576000, 3201186852864000;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2)*x*P(n-1,x) + ((n-1)*x+(10*n-10)*x^2)*d/dx P(n-1,x) + ((n-1)*x^2+(4*n-4)*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2)*x*P[n - 1, x] + ((n - 1)*x + (10*n - 10)*x^2)*D[P[n - 1, x], x] + ((n - 1)*x^2 + (4*n - 4)*x^3)*D[P[n - 1, x], {x, 2}])/(n - 1)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2)*x*v[i - 1] + ((n - 1)*x + (10*n - 10)*x^2)*deriv(v[i - 1], x) + ((n - 1)*x^2 + (4*n - 4)*x^3)*deriv(deriv(v[i - 1], x), x))/(n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-177268549ede7415","steps":[],"edges":[],"text":"A304330"},"headline":"A304330: T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n), triangle read by rows, n >= 0 and 0 <= k <= n.","chain":null}