{"id":"n-deb68900633836aa","revision":1,"first_index":1,"fingerprint":"f6-357ed7a70cf3dab3","name":"basis_map_A008277 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","2"],["15","20","4"],["105","210","84","8"],["945","2520","1512","288","16"],["10395","34650","27720","7920","880","32"],["135135","540540","540540","205920","34320","2496","64"],["2027025","9459450","11351340","5405400","1201200","131040","6720","128"],["34459425","183783600","257297040","147026880","40840800","5940480","456960","17408","256"],["654729075","3928374450","6285399120","4190266080","1396755360","253955520","26046720","1488384","43776","512"],["13749310575","91662070500","164991726900","125707982400","48886437600","10666131840","1367452800","104186880","4596480","107520","1024"],["316234143225","2319050383650","4638100767300","3975514943400","1766895530400","449755225920","69193111680","6589820160","387636480","13601280","259072","2048"],["7905853580625","63246828645000","139143023019000","132517164780000","66258582390000","19275223968000","3459655584000","395389209600","29072736000","1360128000","38860800","614400","4096"],["213458046676875","1849969737866250","4439927370879000","4651352483778000","2584084713210000","845700451596000","173477015712000","23130268761600","2040906067200","119351232000","4546713600","107827200","1437696","8192"],["6190283353629375","57775977967207500","150217542714739500","171677191673988000","104913839356326000","38150487038664000","8803958547384000","1341555588172800","138101310547200","9691320038400","461491430400","14592614400","291852288","3325952","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","39","407","5281","81597","1459015","29582303","669822849","16737058997","457140004519","13541735424615","432232193519521","14783265001765613","539219105249936199"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","2"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166043,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-9009d0d607ac30e3"]},{"id":166044,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-9009d0d607ac30e3"]},{"id":166045,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-9009d0d607ac30e3"]}],"region_exact":{},"incoming_edges":[{"id":"e-9009d0d607ac30e3","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-deb68900633836aa","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"468055c598ec5d72b812a5cc8926a018d5edf1ef560e287d8c414d5f5e537aa8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7c9c430277778c10","role":"sequence","name":"Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-59011d4f00880aa0","kind":"oeis_match","node_a_id":"n-deb68900633836aa","node_b_id":null,"fingerprint":null,"oeis_id":"A223523","in_catalog":0,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":45},"in_catalog":false,"oeis_id":"A223523","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A223523","relation":"match_triangle"}],"bfiles":[{"node_id":"n-deb68900633836aa","oeis_id":"A223523","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":37551}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:         1;\n   2:         3,          2;\n   3:        15,         20,          4;\n   4:       105,        210,         84,          8;\n   5:       945,       2520,       1512,        288,         16;\n   6:     10395,      34650,      27720,       7920,        880,        32;\n   7:    135135,     540540,     540540,     205920,      34320,      2496,       64;\n   8:   2027025,    9459450,   11351340,    5405400,    1201200,    131040,     6720,     128;\n   9:  34459425,  183783600,  257297040,  147026880,   40840800,   5940480,   456960,   17408,   256;\n  10: 654729075, 3928374450, 6285399120, 4190266080, 1396755360, 253955520, 26046720, 1488384, 43776, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n-1+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2*n - 1 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2*n - 1 + 2*x)*v[i - 1] + 2*x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A223523","target":"triangle","first_row":1,"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:         3,          2;\n  2:        15,         20,          4;\n  3:       105,        210,         84,          8;\n  4:       945,       2520,       1512,        288,         16;\n  5:     10395,      34650,      27720,       7920,        880,        32;\n  6:    135135,     540540,     540540,     205920,      34320,      2496,       64;\n  7:   2027025,    9459450,   11351340,    5405400,    1201200,    131040,     6720,     128;\n  8:  34459425,  183783600,  257297040,  147026880,   40840800,   5940480,   456960,   17408,   256;\n  9: 654729075, 3928374450, 6285399120, 4190266080, 1396755360, 253955520, 26046720, 1488384, 43776, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+1+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2*n + 1 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n + 1 + 2*x)*v[i - 1] + 2*x*deriv(v[i - 1], x)); 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":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-7c9c430277778c10","steps":["basis A008277"],"edges":["e-9009d0d607ac30e3"],"text":"A161198 \u2192 basis A008277"},"headline":"A161198 \u2192 basis A008277","chain":null}
