{"id":"n-db27be2f5676b05b","revision":0,"first_index":0,"fingerprint":"f6-7c3b635378939853","name":"theta_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.","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":false,"rows":[["1"],["4","8"],["16","96","48"],["64","832","1248","256"],["256","6272","19968","12544","1280"],["1024","43776","253824","338432","109440","6144"],["4096","291328","2818560","6768640","4697600","873984","28672"],["16384","1879040","28666368","112783360","140979200","57332736","6576640","131072"],["65536","11864064","274427904","1662939136","3383500800","2494408704","640331776","47456256","589824"],["262144","73805824","2515752960","22483853312","69703444480","83644133376","39346743296","6708674560","332126208","2621440"],["1048576","454369280","22340198400","285216604160","1286118400000","2342035734528","1800565760000","570433208320","67020595200","2271846400","11534336"],["4194304","2776547328","193715773440","3447904665600","21850794393600","57481973563392","67062302490624","34961271029760","7757785497600","645719244800","15271010304","50331648"],["16777216","16877387776","1649681891328","40161376010240","348407149363200","1276724788789248","2145993679699968","1702299718385664","627132868853760","100403440025600","6048833601536","101264326656","218103808"],["67108864","102203850752","13855673253888","454403580624896","5285323632640000","26235918627766272","61147119439773696","69882422216884224","39353877941649408","10570647265280000","1249609846718464","55422693015552","664325029888","939524096"],["268435456","617249636352","115137947762688","5024365812383744","77055991010426880","506835404690423808","1590525345684258816","2515767199807832064","2044961158736904192","844725674484039680","169523180222939136","15073097437151232","498931106971648","4320747454464","4026531840"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","12","160","2400","40320","752640","15482880","348364800","8515584000","224811417600","6376469299200","193419568742400","6248939913216000","214249368453120000","7770110429233152000"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","8"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-16","0","0"],["-8","-16","-8"],["0","-8","-12"],["0","0","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-8","0","8"],["0","-4","-8","12"],["0","0","-4","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["8","4"],["6","8"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"],["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188648,"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-081dfcccae930628"]},{"id":188649,"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-081dfcccae930628"]},{"id":188650,"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-081dfcccae930628"]},{"id":188651,"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-081dfcccae930628"]}],"region_exact":{},"incoming_edges":[{"id":"e-081dfcccae930628","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-db27be2f5676b05b","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":"f9d716dd2a2a740c245b54e9a2b43395321ac3ee933431965ab905a07969598a","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-c35e92273dac3e06","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.","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-06c8fc7a44623942","kind":"fingerprint_related","node_a_id":"n-db27be2f5676b05b","node_b_id":"n-5ed42a17b514c663","fingerprint":"f6-7c3b635378939853","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7c3b635378939853"},"other_id":"n-5ed42a17b514c663","other":{"id":"n-5ed42a17b514c663","name":"theta_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.","first_index":0,"fingerprint":"f6-7c3b635378939853","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","12"],["36","216","108"]],"card_rows":[["1"],["6","12"],["36","216","108"],["216","2808","4212","864"]],"row_sums_preview":["1","18","360","8100","204120","5715360","176359680","5952139200"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-61cf3a32da4a0e22","steps":["\u03b8+1"],"edges":["e-fc5203ebe83181fc"],"text":"A257626 \u2192 \u03b8+1"},"headline":"A257626 \u2192 \u03b8+1","chain":null}},{"id":"l-d5597631237d6f0d","kind":"fingerprint_related","node_a_id":"n-ca40f10a3c367093","node_b_id":"n-db27be2f5676b05b","fingerprint":"f6-7c3b635378939853","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7c3b635378939853"},"other_id":"n-ca40f10a3c367093","other":{"id":"n-ca40f10a3c367093","name":"theta_plus_one of Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.","first_index":1,"fingerprint":"f6-7c3b635378939853","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["4","24","12"],["8","104","156","32"],["16","392","1248","784","80"]],"card_rows":[["4","24","12"],["8","104","156","32"],["16","392","1248","784","80"],["32","1368","7932","10576","3420","192"]],"row_sums_preview":["40","300","2520","23520","241920","2721600","33264000","439084800"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9c8a3637306e75dc","steps":["\u03b8+1"],"edges":["e-995d7d6de5709b6d"],"text":"A256890 \u2192 \u03b8+1"},"headline":"A256890 \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,        8;\n  2:     16,       96,         48;\n  3:     64,      832,       1248,         256;\n  4:    256,     6272,      19968,       12544,        1280;\n  5:   1024,    43776,     253824,      338432,      109440,        6144;\n  6:   4096,   291328,    2818560,     6768640,     4697600,      873984,       28672;\n  7:  16384,  1879040,   28666368,   112783360,   140979200,    57332736,     6576640,     131072;\n  8:  65536, 11864064,  274427904,  1662939136,  3383500800,  2494408704,   640331776,   47456256,    589824;\n  9: 262144, 73805824, 2515752960, 22483853312, 69703444480, 83644133376, 39346743296, 6708674560, 332126208, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = -(8*n+16+(8*n^2+16*n)*x+(4*n^3+12*n^2+8*n)*x^2)*P(n-2,x) - ((4*n+8)*x+(4*n^2+8*n)*x^2-(4*n^2+12*n+8)*x^3)*d/dx P(n-2,x) + (6*n+8+(4*n^2+8*n+4)*x)*P(n-1,x) + ((2*n+2)*x-(2*n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n + 16 + (8*n^2 + 16*n)*x + (4*n^3 + 12*n^2 + 8*n)*x^2)*P[n - 2, x] - ((4*n + 8)*x + (4*n^2 + 8*n)*x^2 - (4*n^2 + 12*n + 8)*x^3)*D[P[n - 2, x], x] + (6*n + 8 + (4*n^2 + 8*n + 4)*x)*P[n - 1, x] + ((2*n + 2)*x - (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n + 16 + (8*n^2 + 16*n)*x + (4*n^3 + 12*n^2 + 8*n)*x^2)*v[i - 2] - ((4*n + 8)*x + (4*n^2 + 8*n)*x^2 - (4*n^2 + 12*n + 8)*x^3)*deriv(v[i - 2], x) + (6*n + 8 + (4*n^2 + 8*n + 4)*x)*v[i - 1] + ((2*n + 2)*x - (2*n + 2)*x^2)*deriv(v[i - 1], x))/(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-c35e92273dac3e06","steps":["\u03b8+1"],"edges":["e-081dfcccae930628"],"text":"A257613 \u2192 \u03b8+1"},"headline":"A257613 \u2192 \u03b8+1","chain":null}
