{"id":"n-84fd6a0d7d3c5087","revision":0,"first_index":0,"fingerprint":"f6-eefdbaf8408bdfd8","name":"theta_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","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":true,"rows":[["1"],["12","30"],["144","720","675"],["1728","12960","24300","13500"],["20736","207360","583200","648000","253125"],["248832","3110400","11664000","19440000","15187500","4556250"],["2985984","44789760","209952000","466560000","546750000","328050000","79734375"],["35831808","627056640","3527193600","9797760000","15309000000","13778100000","6697687500","1366875000"],["429981696","8599633920","56435097600","188116992000","367416000000","440899200000","321489000000","131220000000","23066015625"],["5159780352","116095057920","870712934400","3386105856000","7936185600000","11904278400000","11573604000000","7085880000000","2491129687500","384433593750"],["61917364224","1547934105600","13060694016000","58047528960000","158723712000000","285702681600000","347208120000000","283435200000000","149467781250000","46132031250000","6343154296875"],["743008370688","20432730193920","191556845568000","957784227840000","2993075712000000","6285458995200000","9166294368000000","9353361600000000","6576582375000000","3044714062500000","837296367187500","103797070312500"],["8916100448256","267483013447680","2758418576179200","15324547645440000","53875362816000000","129300870758400000","219991064832000000","269376814080000000","236756965500000000","146146275000000000","60285338437500000","14946778125000000","1686702392578125"],["106993205379072","3477279174819840","39119390716723200","239062943268864000","933839622144000000","2521366979788800000","4902658016256000000","7003797166080000000","7386817323600000000","5699704725000000000","3134837598750000000","1165848693750000000","263125573242187500","27246730957031250"],["1283918464548864","44937146259210240","547671470034124800","3651143133560832000","15688505652019200000","47065516956057600000","102955818341376000000","168091131985920000000","206830885060800000000","191510078760000000000","131663179147500000000","65287526850000000000","22102548152343750000","4577450800781250000","437893890380859375"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","42","1539","52488","1712421","54206982","1678822119","51139504548","1537671920841","45753584909922","1349730754842699","39531097362172608","1150725537276993261","33321009035933804862","960455571265497375279"],"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":[["4","0"],["0","5"]]},"terms":[{"coeff":{"coeffs":[["48","120","75"],["0","60","75"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192093,"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-97c057fc15f4e402"]},{"id":192094,"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-97c057fc15f4e402"]},{"id":192095,"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-97c057fc15f4e402"]},{"id":192096,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-97c057fc15f4e402"]}],"region_exact":{},"incoming_edges":[{"id":"e-97c057fc15f4e402","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-84fd6a0d7d3c5087","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":"28959e5a052a253594b88b53454821d3422aa06898b4e8cfeb57af9e2c857292","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-d7df91eaf8843b88","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","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-5cca5e29890790f5","kind":"fingerprint_related","node_a_id":"n-280e93fc5b9e9fe7","node_b_id":"n-84fd6a0d7d3c5087","fingerprint":"f6-eefdbaf8408bdfd8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-eefdbaf8408bdfd8"},"other_id":"n-280e93fc5b9e9fe7","other":{"id":"n-280e93fc5b9e9fe7","name":"theta_plus_one of basis_map_A013609 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-eefdbaf8408bdfd8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","20"],["64","320","300"]],"card_rows":[["1"],["8","20"],["64","320","300"],["512","3840","7200","4000"]],"row_sums_preview":["1","28","684","15552","338256","7138368","147386304","2993075712"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A013609","\u03b8+1"],"edges":["e-c5caeaebd0a50686","e-601435738fe14264"],"text":"A013622 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A013622 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-9822ab2c538dce88","kind":"fingerprint_related","node_a_id":"n-84fd6a0d7d3c5087","node_b_id":"n-a9b3dce6b542bba6","fingerprint":"f6-eefdbaf8408bdfd8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-eefdbaf8408bdfd8"},"other_id":"n-a9b3dce6b542bba6","other":{"id":"n-a9b3dce6b542bba6","name":"theta_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-eefdbaf8408bdfd8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","1280","1200"],["4096","30720","57600","32000"],["65536","655360","1843200","2048000","800000"]],"card_rows":[["256","1280","1200"],["4096","30720","57600","32000"],["65536","655360","1843200","2048000","800000"],["1048576","13107200","49152000","81920000","64000000","19200000"]],"row_sums_preview":["2736","124416","5412096","228427776","9432723456","383113691136","15359376162816","609359740010496"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["basis A013611","\u03b8+1"],"edges":["e-fd91849598d32af1","e-5642bca2af35b2a2"],"text":"A038319 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A038319 \u2192 basis A013611 \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:         12,           30;\n  2:        144,          720,          675;\n  3:       1728,        12960,        24300,         13500;\n  4:      20736,       207360,       583200,        648000,        253125;\n  5:     248832,      3110400,     11664000,      19440000,      15187500,        4556250;\n  6:    2985984,     44789760,    209952000,     466560000,     546750000,      328050000,       79734375;\n  7:   35831808,    627056640,   3527193600,    9797760000,   15309000000,    13778100000,     6697687500,    1366875000;\n  8:  429981696,   8599633920,  56435097600,  188116992000,  367416000000,   440899200000,   321489000000,  131220000000,   23066015625;\n  9: 5159780352, 116095057920, 870712934400, 3386105856000, 7936185600000, 11904278400000, 11573604000000, 7085880000000, 2491129687500, 384433593750;\n","recurrence_sentence":"The row polynomials satisfy (4+5*n*x)*P(n,x) = (48+(60*n+120)*x+(75*n+75)*x^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[((48 + (60*n + 120)*x + (75*n + 75)*x^2)*P[n - 1, x])/(4 + 5*n*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] = ((48 + (60*n + 120)*x + (75*n + 75)*x^2)*v[i - 1])/(4 + 5*n*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-a6a633b989da1879","steps":["basis A013610","\u03b8+1"],"edges":["e-cc1debec4a411a9c","e-97c057fc15f4e402"],"text":"A038271 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A038271 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}