{"id":"n-760706415e8d0c94","revision":0,"first_index":0,"fingerprint":"f6-4f7c344a4c25ac22","name":"theta_plus_one_squared of basis_map_A013609 of Triangle read by rows, A007318 rows repeated.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["2","8"],["2","8"],["4","32","36"],["4","32","36"],["8","96","216","128"],["8","96","216","128"],["16","256","864","1024","400"],["16","256","864","1024","400"],["32","640","2880","5120","4000","1152"],["32","640","2880","5120","4000","1152"],["64","1536","8640","20480","24000","13824","3136"],["64","1536","8640","20480","24000","13824","3136"],["128","3584","24192","71680","112000","96768","43904","8192"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","10","10","72","72","448","448","2560","2560","13824","13824","71680","71680","360448"],"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"]],"recurrence":{"denominator":{"coeffs":[["64","56","4","-12","0"],["0","80","24","-6","0"],["0","0","56","22","1"],["0","0","0","12","2"],["0","0","0","0","1"]]},"terms":[{"coeff":{"coeffs":[["0","-400","-440","-216","-40","-8"],["64","400","-152","-272","-124","-16"],["0","136","596","218","16","-2"],["0","0","80","270","109","14"],["0","0","0","16","42","10"],["0","0","0","0","1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-32","-272","-552","-528","-296","-96","-16"],["0","-96","-632","-1092","-820","-312","-48"],["0","0","-88","-428","-574","-286","-52"],["0","0","0","-24","-92","-92","-24"],["0","0","0","0","-2","-6","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171808,"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-22e8cf39d67b3907"]},{"id":171809,"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-22e8cf39d67b3907"]},{"id":171810,"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-22e8cf39d67b3907"]}],"region_exact":{},"incoming_edges":[{"id":"e-22e8cf39d67b3907","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-760706415e8d0c94","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":"4b0a2f229a1f96d2d9cda8f926cdeb236e8b62170f74af26f20dce7733036c7d","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-b7506a83efd6007f","role":"sequence","name":"basis_map_A013609 of Triangle read by rows, A007318 rows repeated.","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-3c89b58c780c538a","kind":"fingerprint_related","node_a_id":"n-05fe2267b7e2ee62","node_b_id":"n-760706415e8d0c94","fingerprint":"f6-4f7c344a4c25ac22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4f7c344a4c25ac22"},"other_id":"n-05fe2267b7e2ee62","other":{"id":"n-05fe2267b7e2ee62","name":"theta_plus_one_squared of Triangle read by rows, A007318 rows repeated.","first_index":0,"fingerprint":"f6-4f7c344a4c25ac22","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","4"]],"card_rows":[["1"],["1"],["1","4"],["1","4"]],"row_sums_preview":["1","1","5","5","18","18","56","56"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4124f4242e7d4c67","steps":["(\u03b8+1)\u00b2"],"edges":["e-e247b8d27ee67ca0"],"text":"A152198 \u2192 (\u03b8+1)\u00b2"},"headline":"A152198 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-e4ab2d92d4455ba7","kind":"fingerprint_related","node_a_id":"n-760706415e8d0c94","node_b_id":"n-c846bc324cb25f0c","fingerprint":"f6-4f7c344a4c25ac22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4f7c344a4c25ac22"},"other_id":"n-c846bc324cb25f0c","other":{"id":"n-c846bc324cb25f0c","name":"theta_plus_one_squared of derivative of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","first_index":0,"fingerprint":"f6-4f7c344a4c25ac22","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["4"],["8","32"],["16","64"]],"row_sums_preview":["0","0","2","4","40","80","432","864"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b6aca576a689e9f0","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-5c70b541165ff56b","e-6e441004198bccbf"],"text":"A274496 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A274496 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-e4fe9bfc7505d227","kind":"fingerprint_related","node_a_id":"n-44e08c411ecb806d","node_b_id":"n-760706415e8d0c94","fingerprint":"f6-4f7c344a4c25ac22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4f7c344a4c25ac22"},"other_id":"n-44e08c411ecb806d","other":{"id":"n-44e08c411ecb806d","name":"theta_plus_one_squared of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","first_index":0,"fingerprint":"f6-4f7c344a4c25ac22","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["2","8"]],"card_rows":[["1"],["2"],["2","8"],["4","16"]],"row_sums_preview":["1","2","10","20","72","144","448","896"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b6aca576a689e9f0","steps":["(\u03b8+1)\u00b2"],"edges":["e-e6a7da242cad039d"],"text":"A274496 \u2192 (\u03b8+1)\u00b2"},"headline":"A274496 \u2192 (\u03b8+1)\u00b2","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:  1;\n  2:  2,   8;\n  3:  2,   8;\n  4:  4,  32,  36;\n  5:  4,  32,  36;\n  6:  8,  96, 216,  128;\n  7:  8,  96, 216,  128;\n  8: 16, 256, 864, 1024, 400;\n  9: 16, 256, 864, 1024, 400;\n","recurrence_sentence":"The row polynomials satisfy (64+(80*n+56)*x+(56*n^2+24*n+4)*x^2+(12*n^3+22*n^2-6*n-12)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = (64*n+(136*n^2+400*n-400)*x+(80*n^3+596*n^2-152*n-440)*x^2+(16*n^4+270*n^3+218*n^2-272*n-216)*x^3+(n^5+42*n^4+109*n^3+16*n^2-124*n-40)*x^4+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^5)*P(n-1,x) - (32+(96*n+272)*x+(88*n^2+632*n+552)*x^2+(24*n^3+428*n^2+1092*n+528)*x^3+(2*n^4+92*n^3+574*n^2+820*n+296)*x^4+(6*n^4+92*n^3+286*n^2+312*n+96)*x^5+(4*n^4+24*n^3+52*n^2+48*n+16)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((64*n + (136*n^2 + 400*n - 400)*x + (80*n^3 + 596*n^2 - 152*n - 440)*x^2 + (16*n^4 + 270*n^3 + 218*n^2 - 272*n - 216)*x^3 + (n^5 + 42*n^4 + 109*n^3 + 16*n^2 - 124*n - 40)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*P[n - 1, x] - (32 + (96*n + 272)*x + (88*n^2 + 632*n + 552)*x^2 + (24*n^3 + 428*n^2 + 1092*n + 528)*x^3 + (2*n^4 + 92*n^3 + 574*n^2 + 820*n + 296)*x^4 + (6*n^4 + 92*n^3 + 286*n^2 + 312*n + 96)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*D[P[n - 1, x], x])/(64 + (80*n + 56)*x + (56*n^2 + 24*n + 4)*x^2 + (12*n^3 + 22*n^2 - 6*n - 12)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((64*n + (136*n^2 + 400*n - 400)*x + (80*n^3 + 596*n^2 - 152*n - 440)*x^2 + (16*n^4 + 270*n^3 + 218*n^2 - 272*n - 216)*x^3 + (n^5 + 42*n^4 + 109*n^3 + 16*n^2 - 124*n - 40)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*v[i - 1] - (32 + (96*n + 272)*x + (88*n^2 + 632*n + 552)*x^2 + (24*n^3 + 428*n^2 + 1092*n + 528)*x^3 + (2*n^4 + 92*n^3 + 574*n^2 + 820*n + 296)*x^4 + (6*n^4 + 92*n^3 + 286*n^2 + 312*n + 96)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*deriv(v[i - 1], x))/(64 + (80*n + 56)*x + (56*n^2 + 24*n + 4)*x^2 + (12*n^3 + 22*n^2 - 6*n - 12)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)); 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-4124f4242e7d4c67","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-500dd9b78517f374","e-22e8cf39d67b3907"],"text":"A152198 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A152198 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}