{"id":"n-6e2b6743fd2a183f","revision":0,"first_index":0,"fingerprint":"f6-8f766a22b21a7253","name":"theta_plus_one of reversal of Triangle read by rows: A008277 * A007318.","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"],["1","4"],["1","10","15"],["1","18","66","60"],["1","28","183","396","260"],["1","40","405","1540","2355","1218"],["1","54","780","4560","12080","14316","6139"],["1","70","1365","11340","45780","92820","90069","33120"],["1","88","2226","24920","142205","435540","715505","589448","190323"],["1","108","3438","49896","382515","1646316","4052503","5602488","4019580","1159750"],["1","130","5085","92880","921585","5308506","18254320","37540000","44861859","28564570","7464270"],["1","154","7260","163020","2034285","15130962","69348433","198102520","349903719","368725610","211389673","50563164"],["1","180","10065","272580","4180935","39057480","230506738","873937680","2132907084","3303147760","3116762660","1627325376","359377681"],["1","208","13611","437580","8096660","92908530","687436750","3345964336","10793361150","22984638820","31711621084","27119076192","13015078713","2672590508"],["1","238","18018","678496","14909895","206348142","1873387516","11407766656","47093970924","132027735840","249369666546","310417821168","242963186661","107999150798","20744378175"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","26","145","868","5559","37930","274565","2100256","16916595","143013206","1265368801","11687486220","112431224143","1124119019074"],"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","4"]],"recurrence":{"denominator":{"coeffs":[["1","-2"],["1","2"]]},"terms":[{"coeff":{"coeffs":[["-2","0","10","8","-4","0"],["-1","-6","-1","18","12","-4"],["0","-2","-9","-8","5","6"],["0","0","-1","-4","-5","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","2","4","0","-2","0"],["0","0","1","6","8","1","-2"],["0","0","0","1","4","5","2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","0","-5","6"],["2","8","-1","-10"],["0","2","6","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","1","2"],["0","0","-1","-3","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177399,"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-12c9d875448ca32c"]},{"id":177400,"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-12c9d875448ca32c"]},{"id":177401,"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-12c9d875448ca32c"]}],"region_exact":{},"incoming_edges":[{"id":"e-12c9d875448ca32c","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-6e2b6743fd2a183f","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":"c47f60194cc5a27c4dd5b3e95c54ac0bcf218f681be2eda8fd5f3aad6a122402","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-2e7e4df17772fe4d","role":"sequence","name":"reversal of Triangle read by rows: A008277 * A007318.","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,   4;\n  2: 1,  10,   15;\n  3: 1,  18,   66,    60;\n  4: 1,  28,  183,   396,    260;\n  5: 1,  40,  405,  1540,   2355,    1218;\n  6: 1,  54,  780,  4560,  12080,   14316,    6139;\n  7: 1,  70, 1365, 11340,  45780,   92820,   90069,   33120;\n  8: 1,  88, 2226, 24920, 142205,  435540,  715505,  589448,  190323;\n  9: 1, 108, 3438, 49896, 382515, 1646316, 4052503, 5602488, 4019580, 1159750;\n","recurrence_sentence":"The row polynomials satisfy (n+1+(2*n-2)*x)*P(n,x) = -(n+2+(2*n^2+6*n)*x+(n^3+9*n^2+n-10)*x^2+(4*n^3+8*n^2-18*n-8)*x^3+(5*n^3-5*n^2-12*n+4)*x^4+(2*n^3-6*n^2+4*n)*x^5)*P(n-2,x) + ((n+2)*x^2+(n^2+6*n+4)*x^3+(4*n^2+8*n)*x^4+(5*n^2+n-2)*x^5+(2*n^2-2*n)*x^6)*d/dx P(n-2,x) + (2*n+3+(2*n^2+8*n)*x+(6*n^2-n-5)*x^2+(4*n^2-10*n+6)*x^3)*P(n-1,x) - ((n+1)*x^2+(3*n-1)*x^3+(2*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + 2 + (2*n^2 + 6*n)*x + (n^3 + 9*n^2 + n - 10)*x^2 + (4*n^3 + 8*n^2 - 18*n - 8)*x^3 + (5*n^3 - 5*n^2 - 12*n + 4)*x^4 + (2*n^3 - 6*n^2 + 4*n)*x^5)*P[n - 2, x] + ((n + 2)*x^2 + (n^2 + 6*n + 4)*x^3 + (4*n^2 + 8*n)*x^4 + (5*n^2 + n - 2)*x^5 + (2*n^2 - 2*n)*x^6)*D[P[n - 2, x], x] + (2*n + 3 + (2*n^2 + 8*n)*x + (6*n^2 - n - 5)*x^2 + (4*n^2 - 10*n + 6)*x^3)*P[n - 1, x] - ((n + 1)*x^2 + (3*n - 1)*x^3 + (2*n - 2)*x^4)*D[P[n - 1, x], x])/(n + 1 + (2*n - 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 2 + (2*n^2 + 6*n)*x + (n^3 + 9*n^2 + n - 10)*x^2 + (4*n^3 + 8*n^2 - 18*n - 8)*x^3 + (5*n^3 - 5*n^2 - 12*n + 4)*x^4 + (2*n^3 - 6*n^2 + 4*n)*x^5)*v[i - 2] + ((n + 2)*x^2 + (n^2 + 6*n + 4)*x^3 + (4*n^2 + 8*n)*x^4 + (5*n^2 + n - 2)*x^5 + (2*n^2 - 2*n)*x^6)*deriv(v[i - 2], x) + (2*n + 3 + (2*n^2 + 8*n)*x + (6*n^2 - n - 5)*x^2 + (4*n^2 - 10*n + 6)*x^3)*v[i - 1] - ((n + 1)*x^2 + (3*n - 1)*x^3 + (2*n - 2)*x^4)*deriv(v[i - 1], x))/(n + 1 + (2*n - 2)*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-bdae5aa3d1cac0ec","steps":["reverse","\u03b8+1"],"edges":["e-475fdd3e596ddb02","e-12c9d875448ca32c"],"text":"A137597 \u2192 reverse \u2192 \u03b8+1"},"headline":"A137597 \u2192 reverse \u2192 \u03b8+1","chain":null}