{"id":"n-2c096deb92ac6bc1","revision":0,"first_index":0,"fingerprint":"f6-d5c7a04bc225be4d","name":"theta_plus_one_squared of Triangle T(n,k) read by rows; related to number of preorders.","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"],["0","8"],["0","12","54"],["0","0","180","384"],["0","0","135","2080","3000"],["0","0","0","3360","23100","25920"],["0","0","0","1680","59500","263088","246960"],["0","0","0","0","63000","951552","3146976","2580480"],["0","0","0","0","23625","1587600","14879340","39896064","29393280"],["0","0","0","0","0","1247400","34560680","235445760","537593760","362880000"],["0","0","0","0","0","374220","42446250","710321920","3833274456","7699824000","4829932800"],["0","0","0","0","0","0","26486460","1206965760","14363846640","64753603200","117070462080","68976230400"],["0","0","0","0","0","0","6621615","1170449280","31545373860","292052560800","1139692409856","1885930076160","1052366515200"],["0","0","0","0","0","0","0","605404800","42124822740","793492700000","6044296418160","20939123552256","32120196675840","17086945075200"],["0","0","0","0","0","0","0","129729600","33713479800","1364262900000","19702646483520","128243106351360","401816048250240","577084827110400","294226732800000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","66","564","5215","52380","571228","6742008","85779909","1171727600","17116173646","266397594540","4402764006771","77026784648996","1422471467104920"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","8"],["0","12","54"],["0","0","180","384"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-9"],["0","0","0"],["0","0","9"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","-4"],["0","0","0","0"],["0","0","0","4"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","-9"],["0","0","9"],["0","0","18"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","-4"],["0","0","0","4"],["0","0","0","8"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","4","0"],["0","4","9"],["0","0","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4","0"],["0","0","-4","4"],["0","0","0","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","4"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4"],["0","0","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181483,"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-60595bd8a18128ac"]},{"id":181484,"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-60595bd8a18128ac"]},{"id":181485,"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-60595bd8a18128ac"]}],"region_exact":{},"incoming_edges":[{"id":"e-60595bd8a18128ac","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-2c096deb92ac6bc1","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":"cb7fbd2260e0d8f71283a77bb957b70213ea74e442be74671d61192fba471c27","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-f364085508d20047","role":"sequence","name":"Triangle T(n,k) read by rows; related to number of preorders.","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: 0,  8;\n  2: 0, 12,  54;\n  3: 0,  0, 180,  384;\n  4: 0,  0, 135, 2080,  3000;\n  5: 0,  0,   0, 3360, 23100,   25920;\n  6: 0,  0,   0, 1680, 59500,  263088,   246960;\n  7: 0,  0,   0,    0, 63000,  951552,  3146976,   2580480;\n  8: 0,  0,   0,    0, 23625, 1587600, 14879340,  39896064,  29393280;\n  9: 0,  0,   0,    0,     0, 1247400, 34560680, 235445760, 537593760, 362880000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (9*n^2-9)*x^2*P(n-4,x) + (4*n^2-4)*x^3*d/dx P(n-4,x) + (18*n^2+9*n-9)*x^2*P(n-3,x) + (8*n^2+4*n-4)*x^3*d/dx P(n-3,x) + ((4*n+4)*x+(9*n^2+9*n)*x^2)*P(n-2,x) - ((4*n+4)*x^2-(4*n^2+4*n)*x^3)*d/dx P(n-2,x) + (4*n+4)*x*P(n-1,x) - (4*n+4)*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 8*x, P(2,x) = 12*x + 54*x^2, P(3,x) = 180*x^2 + 384*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 8*x;\nP[2, x_] := 12*x + 54*x^2;\nP[3, x_] := 180*x^2 + 384*x^3;\nP[n_, x_] := P[n, x] = Expand[(9*n^2 - 9)*x^2*P[n - 4, x] + (4*n^2 - 4)*x^3*D[P[n - 4, x], x] + (18*n^2 + 9*n - 9)*x^2*P[n - 3, x] + (8*n^2 + 4*n - 4)*x^3*D[P[n - 3, x], x] + ((4*n + 4)*x + (9*n^2 + 9*n)*x^2)*P[n - 2, x] - ((4*n + 4)*x^2 - (4*n^2 + 4*n)*x^3)*D[P[n - 2, x], x] + (4*n + 4)*x*P[n - 1, x] - (4*n + 4)*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 8*x; v[3] = 12*x + 54*x^2; v[4] = 180*x^2 + 384*x^3; for(n = 4, N, my(i = n + 1); v[i] = (9*n^2 - 9)*x^2*v[i - 4] + (4*n^2 - 4)*x^3*deriv(v[i - 4], x) + (18*n^2 + 9*n - 9)*x^2*v[i - 3] + (8*n^2 + 4*n - 4)*x^3*deriv(v[i - 3], x) + ((4*n + 4)*x + (9*n^2 + 9*n)*x^2)*v[i - 2] - ((4*n + 4)*x^2 - (4*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) + (4*n + 4)*x*v[i - 1] - (4*n + 4)*x^2*deriv(v[i - 1], 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-f364085508d20047","steps":["(\u03b8+1)\u00b2"],"edges":["e-60595bd8a18128ac"],"text":"A079510 \u2192 (\u03b8+1)\u00b2"},"headline":"A079510 \u2192 (\u03b8+1)\u00b2","chain":null}