{"id":"n-8854f7202c6a83aa","revision":0,"first_index":0,"fingerprint":"f6-6c0433a6238a5048","name":"theta_plus_one of A decomposition of the double factorials A001147.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1","4"],["1","20","12"],["1","72","180","32"],["1","232","1548","1184","80"],["1","716","10524","20672","6640","192"],["1","2172","63360","256960","214800","33984","448"],["1","6544","355896","2643520","4502800","1920384","164416","1024"],["1","19664","1916904","24198976","73569200","64351104","15580096","766976","2304"],["1","59028","10056456","205175040","1025107440","1590692544","803151552","118339584","3490560","5120"],["1","177124","51876036","1650763008","12816530160","32259165120","28999049856","9118193664","858131712","15610880","11264"],["1","531416","264693900","12800365920","148364031600","570168523008","828615959808","468190617600","96651982080","6017484800","68924416","24576"],["1","1594296","1341006228","96661065440","1623284517600","9120443159808","20084760019968","18386735284224","6910228143360","973460940800","41162812416","301326336","53248"],["1","4782940","6762425796","715963640768","17026087294400","135387820338048","431350152576000","597157189582848","365694804774144","95276070579200","9430875339776","276359823360","1307078656","114688"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","5","33","285","3045","38745","571725","9594585","180405225","3756077325","85769508825","2131143139125","57238379923725","1652323398350625"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["1","-8","-8"],["0","4","12"],["0","0","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0"],["0","2","2","-4"],["0","0","-4","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","4"],["0","-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":180711,"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-0a7d24734295b6da"]},{"id":180712,"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-0a7d24734295b6da"]},{"id":180713,"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-0a7d24734295b6da"]},{"id":180714,"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-0a7d24734295b6da"]}],"region_exact":{},"incoming_edges":[{"id":"e-0a7d24734295b6da","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-8854f7202c6a83aa","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":"e7994dc691d7b225f23b0490963f4d2bd5fa68c5f8eb0a710223409d0ffb4a03","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-54b58a0ddb0e34fe","role":"sequence","name":"A decomposition of the double factorials A001147.","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":[],"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: 1;\n  2: 1,     4;\n  3: 1,    20,      12;\n  4: 1,    72,     180,       32;\n  5: 1,   232,    1548,     1184,       80;\n  6: 1,   716,   10524,    20672,     6640,      192;\n  7: 1,  2172,   63360,   256960,   214800,    33984,      448;\n  8: 1,  6544,  355896,  2643520,  4502800,  1920384,   164416,   1024;\n  9: 1, 19664, 1916904, 24198976, 73569200, 64351104, 15580096, 766976, 2304;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (n+(4*n^2-8*n)*x-(4*n^3-12*n^2+8*n)*x^2)*P(n-2,x) + (2*n*x-(4*n^2-2*n)*x^2+(4*n^2-4*n)*x^3)*d/dx P(n-2,x) - (1-(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) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n + (4*n^2 - 8*n)*x - (4*n^3 - 12*n^2 + 8*n)*x^2)*P[n - 2, x] + (2*n*x - (4*n^2 - 2*n)*x^2 + (4*n^2 - 4*n)*x^3)*D[P[n - 2, x], x] - (1 - (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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n + (4*n^2 - 8*n)*x - (4*n^3 - 12*n^2 + 8*n)*x^2)*v[i - 2] + (2*n*x - (4*n^2 - 2*n)*x^2 + (4*n^2 - 4*n)*x^3)*deriv(v[i - 2], x) - (1 - (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-54b58a0ddb0e34fe","steps":["\u03b8+1"],"edges":["e-0a7d24734295b6da"],"text":"A185410 \u2192 \u03b8+1"},"headline":"A185410 \u2192 \u03b8+1","chain":null}