{"id":"n-f58672b7a8bd976b","revision":0,"first_index":0,"fingerprint":"f6-e422756c786abb1e","name":"theta_plus_one of basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["36","240","300"],["216","2160","5400","4000"],["1296","17280","64800","96000","50000"],["7776","129600","648000","1440000","1500000","600000"],["46656","933120","5832000","17280000","27000000","21600000","7000000"],["279936","6531840","48988800","181440000","378000000","453600000","294000000","80000000"],["1679616","44789760","391910400","1741824000","4536000000","7257600000","7056000000","3840000000","900000000"],["10077696","302330880","3023308800","15676416000","48988800000","97977600000","127008000000","103680000000","48600000000","10000000000"],["60466176","2015539200","22674816000","134369280000","489888000000","1175731200000","1905120000000","2073600000000","1458000000000","600000000000","110000000000"],["362797056","13302558720","166281984000","1108546560000","4618944000000","12933043200000","25147584000000","34214400000000","32076000000000","19800000000000","7260000000000","1200000000000"],["2176782336","87071293440","1197230284800","8868372480000","41570496000000","133025587200000","301771008000000","492687360000000","577368000000000","475200000000000","261360000000000","86400000000000","13000000000000"],["13060694016","565963407360","8489451110400","69173305344000","360277632000000","1296999475200000","3362591232000000","6404935680000000","9006940800000000","9266400000000000","6795360000000000","3369600000000000","1014000000000000","140000000000000"],["78364164096","3656994324480","59426157772800","528232513536000","3026332108800000","12105328435200000","35307207936000000","76859228160000000","126097171200000000","155675520000000000","142702560000000000","94348800000000000","42588000000000000","11760000000000000","1500000000000000"],["470184984576","23509249228800","411411861504000","3961743851520000","24760899072000000","108947955916800000","353072079360000000","864666316800000000","1621249344000000000","2335132800000000000","2568646080000000000","2122848000000000000","1277640000000000000","529200000000000000","135000000000000000","16000000000000000"],["2821109907456","150459195064320","2821109907456000","29255954595840000","198087192576000000","950818524364800000","3389491961856000000","9223107379200000000","19454992128000000000","32024678400000000000","41098337280000000000","40758681600000000000","30663360000000000000","16934400000000000000","6480000000000000000","1536000000000000000","170000000000000000"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["576","11776","229376","4325376","79691776","1442840576","25769803776","455266533376","7971459301376","138538465099776","2392537302040576","41095346599755776","702561541869797376","11961560610296037376","202914184810805067776"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["36","240","300"]],"recurrence":{"denominator":{"coeffs":[["3","10"],["0","5"]]},"terms":[{"coeff":{"coeffs":[["18","120","150"],["0","30","50"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192435,"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-0463688828a3a0c7"]},{"id":192436,"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-0463688828a3a0c7"]},{"id":192437,"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-0463688828a3a0c7"]},{"id":192438,"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-0463688828a3a0c7"]}],"region_exact":{},"incoming_edges":[{"id":"e-0463688828a3a0c7","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-f58672b7a8bd976b","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":"d0a7b7db3140e1621c9fc5384b3a4fd419c6a5f3786b5f405bf4c738dc68a886","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-d524cbe046c3619e","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.","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-350294d9389a0870","kind":"fingerprint_related","node_a_id":"n-cd2143df0e79ad41","node_b_id":"n-f58672b7a8bd976b","fingerprint":"f6-e422756c786abb1e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e422756c786abb1e"},"other_id":"n-cd2143df0e79ad41","other":{"id":"n-cd2143df0e79ad41","name":"theta_plus_one of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-e422756c786abb1e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","10"],["9","60","75"]],"card_rows":[["1"],["3","10"],["9","60","75"],["27","270","675","500"]],"row_sums_preview":["1","13","144","1472","14336","135168","1245184","11272192"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["\u03b8+1"],"edges":["e-a296c99492627be0"],"text":"A013622 \u2192 \u03b8+1"},"headline":"A013622 \u2192 \u03b8+1","chain":null}},{"id":"l-4a36f90ddc0a0a95","kind":"fingerprint_related","node_a_id":"n-f58672b7a8bd976b","node_b_id":"n-35a5ff45224c6951","fingerprint":"f6-e422756c786abb1e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e422756c786abb1e"},"other_id":"n-35a5ff45224c6951","other":{"id":"n-35a5ff45224c6951","name":"theta_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-e422756c786abb1e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","40"],["144","960","1200"]],"card_rows":[["1"],["12","40"],["144","960","1200"],["1728","17280","43200","32000"]],"row_sums_preview":["1","52","2304","94208","3670016","138412032","5100273664","184683593728"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013611","\u03b8+1"],"edges":["e-62a1b0a86f60f435","e-3eeb8555e7c2d921"],"text":"A038271 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A038271 \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:        36,         240,          300;\n  1:       216,        2160,         5400,          4000;\n  2:      1296,       17280,        64800,         96000,         50000;\n  3:      7776,      129600,       648000,       1440000,       1500000,         600000;\n  4:     46656,      933120,      5832000,      17280000,      27000000,       21600000,        7000000;\n  5:    279936,     6531840,     48988800,     181440000,     378000000,      453600000,      294000000,       80000000;\n  6:   1679616,    44789760,    391910400,    1741824000,    4536000000,     7257600000,     7056000000,     3840000000,      900000000;\n  7:  10077696,   302330880,   3023308800,   15676416000,   48988800000,    97977600000,   127008000000,   103680000000,    48600000000,    10000000000;\n  8:  60466176,  2015539200,  22674816000,  134369280000,  489888000000,  1175731200000,  1905120000000,  2073600000000,  1458000000000,   600000000000,  110000000000;\n  9: 362797056, 13302558720, 166281984000, 1108546560000, 4618944000000, 12933043200000, 25147584000000, 34214400000000, 32076000000000, 19800000000000, 7260000000000, 1200000000000;\n","recurrence_sentence":"The row polynomials satisfy (3+(5*n+10)*x)*P(n,x) = (18+(30*n+120)*x+(50*n+150)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 240*x + 300*x^2.","mathematica":"P[0, x_] := 36 + 240*x + 300*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((18 + (30*n + 120)*x + (50*n + 150)*x^2)*P[n - 1, x])/(3 + (5*n + 10)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 36 + 240*x + 300*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((18 + (30*n + 120)*x + (50*n + 150)*x^2)*v[i - 1])/(3 + (5*n + 10)*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-bbfc5a54ff13a785","steps":["basis A013609","\u03b8+1"],"edges":["e-31086658f6c15e80","e-0463688828a3a0c7"],"text":"A013612 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A013612 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}