{"id":"n-3e9fa36c1d067fb0","revision":0,"first_index":0,"fingerprint":"f6-875f7e3274313d29","name":"theta_plus_one of basis_map_A013610 of Even-numbered rows of Pascal's triangle.","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":true,"rows":[["1"],["4","24","27"],["16","192","648","864","405"],["64","1152","6480","17280","24300","17496","5103"],["256","6144","48384","193536","453600","653184","571536","279936","59049"],["1024","30720","311040","1658880","5443200","11757312","17146080","16796160","10628820","3936600","649539"],["4096","147456","1824768","12165120","51321600","147806208","301771008","443418624","467668080","346420800","171478296","51018336","6908733"],["16384","688128","10063872","80510976","415134720","1494484992","3923023104","7685922816","11348745408","12609717120","10403016624","6190224768","2514778812","624974616","71744535"],["65536","3145728","53084160","495452160","3019161600","13042778112","41845579776","102478970880","194549921280","288222105600","332896531968","297130788864","201182304960","99995938560","34437376800","7346640384","731794257"],["262144","14155776","270729216","2887778304","20304691200","102335643648","388022648832","1140311457792","2645878929408","4899775795200","7276167055872","8659240132608","8208238042368","6119751439872","3512612433600","1498714638336","447858085284","83682825624","7360989291"],["1048576","62914560","1344798720","16137584640","128596377600","740715134976","3240628715520","11110727024640","30467696762880","67705992806400","122886376943616","182806180577280","222795032578560","221476718776320","177972363302400","113902312513536","56728690802640","21199649158080","5594351861160","929809173600","73222472421"],["4194304","276824064","6539968512","87199580160","776621260800","5032505769984","24952841109504","97774397816832","309364305592320","802055607091200","1720409277210624","3071143833698304","4574724668946432","5684569115258880","5873087988979200","5011701750595584","3494487353442624","1958847582206592","861530186618640","286381225468800","67657564517004","10125621900504","721764371007"],["16777216","1207959552","31255953408","458420649984","4512578273280","32490563567616","180051873103872","793698052866048","2846151611449344","8433041811701760","20871778483961856","43468497338499072","76522667189649408","114104805513560064","144086425329623040","153692187018264576","137782644221452032","102979415750289408","63408621735131904","31616487291755520","12448991871128736","3726228859385472","796827865591728","108452942008704","7060738412025"],["67108864","5234491392","147220070400","2355521126400","25395462144000","201132060180480","1231933868605440","6033961805414400","24182987548262400","80609958494208000","226110933576253440","538181395619512320","1093180959852134400","1901746758559334400","2838065953462272000","3632724420431708160","3980387499730836480","3718701124316006400","2943971723416838400","1957211118061056000","1078912628831157120","484409751720111360","172646037544874400","46996274870438400","9178959935632500","1145534199966936","68630377364883"],["268435456","22548578304","684913065984","11871826477056","139122966528000","1202022430801920","8063567139962880","43444524998983680","192445669331435520","712761738264576000","2234508049459445760","5983310810122813440","13774080094136893440","27385153323254415360","47155557380603904000","70418965688368496640","91187059084742799360","102230474544832757760","98917449906805770240","82202866958564352000","58261281956882484480","34877502123848017920","17402720584523339520","7105836760410286080","2313097903779390000","577349236783335744","103769130575703096","11956936856459616","663426981193869"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","55","2125","71875","2265625","68359375","2001953125","57373046875","1617431640625","45013427734375","1239776611328125","33855438232421875","917911529541015625","24735927581787109375","663101673126220703125"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2","-3"],["0","6"]]},"terms":[{"coeff":{"coeffs":[["8","36","54","27"],["0","24","72","54"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191306,"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-69b162cf33fe7a77"]},{"id":191307,"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-69b162cf33fe7a77"]},{"id":191308,"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-69b162cf33fe7a77"]}],"region_exact":{},"incoming_edges":[{"id":"e-69b162cf33fe7a77","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-3e9fa36c1d067fb0","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"3699f97fac4786f04f61d851b6710ea1e3ed2c0eaeffeb0a4b8a47bde776cfae","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-1e083728b424e77f","role":"sequence","name":"basis_map_A013610 of Even-numbered rows of Pascal's triangle.","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-0b3bddd71cefa5fe","kind":"fingerprint_related","node_a_id":"n-3e9fa36c1d067fb0","node_b_id":"n-aef5b6c97c5c498f","fingerprint":"f6-875f7e3274313d29","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-875f7e3274313d29"},"other_id":"n-aef5b6c97c5c498f","other":{"id":"n-aef5b6c97c5c498f","name":"theta_plus_one of basis_map_A013610 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","first_index":0,"fingerprint":"f6-875f7e3274313d29","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","48","54"],["64","768","2592","3456","1620"]],"card_rows":[["1"],["8","48","54"],["64","768","2592","3456","1620"],["512","9216","51840","138240","194400","139968","40824"]],"row_sums_preview":["1","110","8500","575000","36250000","2187500000","128125000000","7343750000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["basis A013610","\u03b8+1"],"edges":["e-a142fb8711dcbce4","e-bfc5297eae6f6917"],"text":"A139548 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A139548 \u2192 basis A013610 \u2192 \u03b8+1","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:      4,       24,        27;\n  2:     16,      192,       648,        864,         405;\n  3:     64,     1152,      6480,      17280,       24300,        17496,         5103;\n  4:    256,     6144,     48384,     193536,      453600,       653184,       571536,        279936,         59049;\n  5:   1024,    30720,    311040,    1658880,     5443200,     11757312,     17146080,      16796160,      10628820,       3936600,        649539;\n  6:   4096,   147456,   1824768,   12165120,    51321600,    147806208,    301771008,     443418624,     467668080,     346420800,     171478296,      51018336,       6908733;\n  7:  16384,   688128,  10063872,   80510976,   415134720,   1494484992,   3923023104,    7685922816,   11348745408,   12609717120,   10403016624,    6190224768,    2514778812,     624974616,      71744535;\n  8:  65536,  3145728,  53084160,  495452160,  3019161600,  13042778112,  41845579776,  102478970880,  194549921280,  288222105600,  332896531968,  297130788864,  201182304960,   99995938560,   34437376800,    7346640384,    731794257;\n  9: 262144, 14155776, 270729216, 2887778304, 20304691200, 102335643648, 388022648832, 1140311457792, 2645878929408, 4899775795200, 7276167055872, 8659240132608, 8208238042368, 6119751439872, 3512612433600, 1498714638336, 447858085284, 83682825624, 7360989291;\n","recurrence_sentence":"The row polynomials satisfy (2+(6*n-3)*x)*P(n,x) = (8+(24*n+36)*x+(72*n+54)*x^2+(54*n+27)*x^3)*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[((8 + (24*n + 36)*x + (72*n + 54)*x^2 + (54*n + 27)*x^3)*P[n - 1, x])/(2 + (6*n - 3)*x)]];\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] = ((8 + (24*n + 36)*x + (72*n + 54)*x^2 + (54*n + 27)*x^3)*v[i - 1])/(2 + (6*n - 3)*x)); 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-08f49539099949d5","steps":["basis A013610","\u03b8+1"],"edges":["e-cb7f9b19b94bd36e","e-69b162cf33fe7a77"],"text":"A034870 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A034870 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}