{"id":"n-03edc5c28196c3db","revision":0,"first_index":0,"fingerprint":"f6-6a8026fb20183be6","name":"basis_map_A008277 of theta_plus_one of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","9","3"],["5","75","125","50","5"],["7","441","2107","2450","980","147","7"],["9","2295","27225","69930","62559","23814","4158","324","9"],["11","11253","313511","1603250","2714030","1974357","703857","130680","12705","605","11"],["13","53235","3401125","32922890","97610513","121177056","74300512","24694956","4673526","511225","31603","1014","13"],["15","245745","35626515","635339250","3161503800","6310399095","6131119995","3249417600","1006927350","189939750","22192170","1597050","68250","1575","15"],["17","1114095","364783025","11803733930","96096778067","297597748266","437037781362","347071915476","161989979151","46891680550","8705036626","1054379508","83473026","4248300","132940","2312","17"],["19","4980717","3675513919","213933672050","2804196409490","13168550433801","28365568061941","32485269066120","21747677909295","9068643641915","2458851138029","445872074700","54956344040","4627973070","264418782","10015584","239343","3249","19"],["21","22020075","36589216125","3811690471050","79616513936421","558299268718884","1728530101513908","2782731322288764","2588722005769284","1494929778116775","563363885469021","143493882633738","25303093584951","3135384094500","274836714180","17008833744","734008779","21492135","404985","4410","21"],["23","96468969","360785716523","67075879219250","2217588428234300","22976306733618315","100800765979698015","224064965457709200","283361582671328700","220648229838209580","111877780105875300","38459743790130390","9229498887851970","1578471043563645","195026301874613","17534305927072","1148090742469","54392363487","1834980917","42769650","651728","5819","23"],["25","419430375","3529949775625","1169282243470250","60917124352768775","925660425000060750","5695812074967907750","17255593027959214500","29198036277324325125","30079084804384687500","20058897610961566500","9056565519621867000","2862126833593606500","648952759022400000","107484866383680000","13166379042399000","1201583284827750","81891964873125","4155474241875","155504868750","4212987625","80011250","1006250","7500","25"],["27","1811939301","34313376743127","20231884040454450","1656131593277045430","36714635586136202877","314484437868603521337","1285978458966021077400","2877208378574628480525","3866320903754437353225","3335024262343412493855","1939225497856613530200","788586268117133959200","230611029258642813900","49535119924096903020","7939702783912273920","960371579625912090","88295173880683770","6190249161366270","330650000430750","13380229515510","405419884155","9013464135","142155000","1500525","9477","27"],["29","7784628195","331705705968725","347946661585035530","44665811924012670329","1439221194622916536104","17065623394202369874888","93505239805399096345164","274135922601490377556674","475369104177500119569125","523726985552894482036535","386473709934857515509630","198796869985039900213185","73499765839988275431000","20001093884690658648600","4080153560297351074080","632729194475837983530","75357406956913385850","6940637624202541350","496195253668214100","27547401165394110","1183869248604900","39087968817300","978410569500","18169990020","241761429","2172303","11774","29"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","15","260","6139","190323","7464270","359377681","20744378175","1408702786668","110822101896083","9972266139291771","1015496134666939958","115964758305749983825","14734360294273619713503","2068854256226947980543988"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["2"]]},"terms":[{"coeff":{"coeffs":[["1","3","1"],["2","6","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","2"],["0","6","4"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"],["0","0","2"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195859,"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-df664ce7639b6c95"]},{"id":195860,"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-df664ce7639b6c95"]},{"id":195861,"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-df664ce7639b6c95"]}],"region_exact":{},"incoming_edges":[{"id":"e-df664ce7639b6c95","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-03edc5c28196c3db","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":"4ee76bc03b18b3ea8b1c3666eb333852931dd8977ff927b06bfed230065f56dc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f3766a15d78768ea","role":"sequence","name":"theta_plus_one of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-13b1461ec7bbd412","kind":"fingerprint_related","node_a_id":"n-03edc5c28196c3db","node_b_id":"n-83e65651eddead61","fingerprint":"f6-6a8026fb20183be6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6a8026fb20183be6"},"other_id":"n-83e65651eddead61","other":{"id":"n-83e65651eddead61","name":"basis_map_A008277 of theta_plus_one_squared of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-6a8026fb20183be6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["9","27","9"],["25","375","625","250","25"]],"card_rows":[["1"],["9","27","9"],["25","375","625","250","25"],["49","3087","14749","17150","6860","1029","49"]],"row_sums_preview":["1","45","1300","42973","1712907","82106970","4671909853","311165672625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["(\u03b8+1)\u00b2","basis A008277"],"edges":["e-5ac7ff393747d6fc","e-43c09f07c4e1e68b"],"text":"basis-map coefficients of A297191, shifted \u2192 (\u03b8+1)\u00b2 \u2192 basis A008277"},"headline":"basis-map coefficients of A297191, shifted \u2192 (\u03b8+1)\u00b2 \u2192 basis A008277","chain":null}},{"id":"l-3cf7f6ce9aaceac0","kind":"fingerprint_related","node_a_id":"n-64a8f6054296172d","node_b_id":"n-03edc5c28196c3db","fingerprint":"f6-6a8026fb20183be6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6a8026fb20183be6"},"other_id":"n-64a8f6054296172d","other":{"id":"n-64a8f6054296172d","name":"basis_map_A008277 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-6a8026fb20183be6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3","1"],["1","15","25","10","1"]],"card_rows":[["1"],["1","3","1"],["1","15","25","10","1"],["1","63","301","350","140","21","1"]],"row_sums_preview":["1","5","52","877","21147","678570","27644437","1382958545"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["basis A008277"],"edges":["e-7ccdf7ec12150e18"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A008277"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A008277","chain":null}},{"id":"l-f36c2ac192551f7d","kind":"fingerprint_related","node_a_id":"n-cc5ac69d87f94be7","node_b_id":"n-03edc5c28196c3db","fingerprint":"f6-6a8026fb20183be6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6a8026fb20183be6"},"other_id":"n-cc5ac69d87f94be7","other":{"id":"n-cc5ac69d87f94be7","name":"reversal of basis_map_A008277 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-6a8026fb20183be6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3","1"],["1","10","25","15","1"]],"card_rows":[["1"],["1","3","1"],["1","10","25","15","1"],["1","21","140","350","301","63","1"]],"row_sums_preview":["1","5","52","877","21147","678570","27644437","1382958545"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["basis A008277","reverse"],"edges":["e-7ccdf7ec12150e18","e-44293a9d609f948d"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A008277 \u2192 reverse"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A008277 \u2192 reverse","chain":null}}],"oeis":[],"bfiles":[],"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:  3,       9,          3;\n  2:  5,      75,        125,           50,             5;\n  3:  7,     441,       2107,         2450,           980,            147,              7;\n  4:  9,    2295,      27225,        69930,         62559,          23814,           4158,            324,              9;\n  5: 11,   11253,     313511,      1603250,       2714030,        1974357,         703857,         130680,          12705,           605,            11;\n  6: 13,   53235,    3401125,     32922890,      97610513,      121177056,       74300512,       24694956,        4673526,        511225,         31603,         1014,          13;\n  7: 15,  245745,   35626515,    635339250,    3161503800,     6310399095,     6131119995,     3249417600,     1006927350,     189939750,      22192170,      1597050,       68250,       1575,        15;\n  8: 17, 1114095,  364783025,  11803733930,   96096778067,   297597748266,   437037781362,   347071915476,   161989979151,   46891680550,    8705036626,   1054379508,    83473026,    4248300,    132940,     2312,     17;\n  9: 19, 4980717, 3675513919, 213933672050, 2804196409490, 13168550433801, 28365568061941, 32485269066120, 21747677909295, 9068643641915, 2458851138029, 445872074700, 54956344040, 4627973070, 264418782, 10015584, 239343, 3249, 19;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+(6*n+3)*x+(2*n+1)*x^2)*P(n-1,x) + ((6*n+3)*x+(4*n+2)*x^2)*d/dx P(n-1,x) + (2*n+1)*x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 1 + (6*n + 3)*x + (2*n + 1)*x^2)*P[n - 1, x] + ((6*n + 3)*x + (4*n + 2)*x^2)*D[P[n - 1, x], x] + (2*n + 1)*x^2*D[P[n - 1, x], {x, 2}])/(2*n - 1)]];\nTable[CoefficientList[P[n, x], x], {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 + (6*n + 3)*x + (2*n + 1)*x^2)*v[i - 1] + ((6*n + 3)*x + (4*n + 2)*x^2)*deriv(v[i - 1], x) + (2*n + 1)*x^2*deriv(deriv(v[i - 1], x), x))/(2*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}}],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["\u03b8+1","basis A008277"],"edges":["e-5ff024f1a04a7dc8","e-df664ce7639b6c95"],"text":"basis-map coefficients of A297191, shifted \u2192 \u03b8+1 \u2192 basis A008277"},"headline":"basis-map coefficients of A297191, shifted \u2192 \u03b8+1 \u2192 basis A008277","chain":null}
