{"id":"n-6cfb5fe6d2148054","revision":0,"first_index":0,"fingerprint":"f6-4a9146a6908256f6","name":"theta_plus_one of basis_map_A013611 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).","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":true,"rows":[["1"],["2"],["4","16"],["8","32"],["16","128","192"],["32","256","384"],["64","768","2304","2048"],["128","1536","4608","4096"],["256","4096","18432","32768","20480"],["512","8192","36864","65536","40960"],["1024","20480","122880","327680","409600","196608"],["2048","40960","245760","655360","819200","393216"],["4096","98304","737280","2621440","4915200","4718592","1835008"],["8192","196608","1474560","5242880","9830400","9437184","3670016"],["16384","458752","4128768","18350080","45875200","66060288","51380224","16777216"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","20","40","336","672","5184","10368","76032","152064","1078272","2156544","14929920","29859840","203046912"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2","3","0"],["0","2","1"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["4","-10","-24","-8"],["0","20","10","-4"],["0","0","18","8"],["0","0","0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-8","-32","-40","-16"],["0","0","-12","-36","-24"],["0","0","0","-4","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175558,"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-3de5ce8dcab4eeb0"]},{"id":175559,"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-3de5ce8dcab4eeb0"]},{"id":175560,"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-3de5ce8dcab4eeb0"]}],"region_exact":{},"incoming_edges":[{"id":"e-3de5ce8dcab4eeb0","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-6cfb5fe6d2148054","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":"44e8f312087517c2065fdce34bcf0681eeac3fbfd34c1b0c1875c5565f5d2a1a","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-98c21bd2a6c7c1d4","role":"sequence","name":"basis_map_A013611 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).","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-5d439b60767c7bc9","kind":"fingerprint_related","node_a_id":"n-5265efb6991612b1","node_b_id":"n-6cfb5fe6d2148054","fingerprint":"f6-4a9146a6908256f6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4a9146a6908256f6"},"other_id":"n-5265efb6991612b1","other":{"id":"n-5265efb6991612b1","name":"theta_plus_one of Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","first_index":0,"fingerprint":"f6-4a9146a6908256f6","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["1","4"]],"card_rows":[["1"],["1"],["1","4"],["1","4"]],"row_sums_preview":["1","1","5","5","21","21","81","81"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ba3bb18490b15e2e","steps":["\u03b8+1"],"edges":["e-f85b46c82d805226"],"text":"A188440 \u2192 \u03b8+1"},"headline":"A188440 \u2192 \u03b8+1","chain":null}},{"id":"l-b202037a8b8b69fb","kind":"fingerprint_related","node_a_id":"n-811069444ca2f8b3","node_b_id":"n-6cfb5fe6d2148054","fingerprint":"f6-4a9146a6908256f6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4a9146a6908256f6"},"other_id":"n-811069444ca2f8b3","other":{"id":"n-811069444ca2f8b3","name":"theta_plus_one of basis_map_A013611 of Triangle read by rows, A007318 rows repeated.","first_index":0,"fingerprint":"f6-4a9146a6908256f6","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["2","8"]],"card_rows":[["1"],["1"],["2","8"],["2","8"]],"row_sums_preview":["1","1","10","10","84","84","648","648"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4124f4242e7d4c67","steps":["basis A013611","\u03b8+1"],"edges":["e-0bf2c6340da57ed0","e-40d1df77ac9e99b0"],"text":"A152198 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A152198 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}},{"id":"l-c5d2e4047fdbee83","kind":"fingerprint_related","node_a_id":"n-1971eb333384ff5d","node_b_id":"n-6cfb5fe6d2148054","fingerprint":"f6-4a9146a6908256f6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4a9146a6908256f6"},"other_id":"n-1971eb333384ff5d","other":{"id":"n-1971eb333384ff5d","name":"theta_plus_one of derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).","first_index":0,"fingerprint":"f6-4a9146a6908256f6","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["6"]],"card_rows":[["6"],["18"],["36","144"],["108","432"]],"row_sums_preview":["0","0","6","18","180","540","3402","10206"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7c641d81e21514ce","steps":["D","\u03b8+1"],"edges":["e-7b8439e885873d7d","e-2ca1462941c186d1"],"text":"A274498 \u2192 D \u2192 \u03b8+1"},"headline":"A274498 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-f1b1182fb2801828","kind":"fingerprint_related","node_a_id":"n-5bbc9a303e54ec16","node_b_id":"n-6cfb5fe6d2148054","fingerprint":"f6-4a9146a6908256f6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4a9146a6908256f6"},"other_id":"n-5bbc9a303e54ec16","other":{"id":"n-5bbc9a303e54ec16","name":"theta_plus_one of Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).","first_index":0,"fingerprint":"f6-4a9146a6908256f6","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3"],["3","12"]],"card_rows":[["1"],["3"],["3","12"],["9","36"]],"row_sums_preview":["1","3","15","45","189","567","2187","6561"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7c641d81e21514ce","steps":["\u03b8+1"],"edges":["e-493e6c3c99ee8db4"],"text":"A274498 \u2192 \u03b8+1"},"headline":"A274498 \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:   2;\n  2:   4,   16;\n  3:   8,   32;\n  4:  16,  128,   192;\n  5:  32,  256,   384;\n  6:  64,  768,  2304,  2048;\n  7: 128, 1536,  4608,  4096;\n  8: 256, 4096, 18432, 32768, 20480;\n  9: 512, 8192, 36864, 65536, 40960;\n","recurrence_sentence":"The row polynomials satisfy (2+(2*n+3)*x+(n^2+n)*x^2)*P(n,x) = (4+(20*n-10)*x+(18*n^2+10*n-24)*x^2+(4*n^3+8*n^2-4*n-8)*x^3)*P(n-1,x) - (8*x+(12*n+32)*x^2+(4*n^2+36*n+40)*x^3+(8*n^2+24*n+16)*x^4)*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[((4 + (20*n - 10)*x + (18*n^2 + 10*n - 24)*x^2 + (4*n^3 + 8*n^2 - 4*n - 8)*x^3)*P[n - 1, x] - (8*x + (12*n + 32)*x^2 + (4*n^2 + 36*n + 40)*x^3 + (8*n^2 + 24*n + 16)*x^4)*D[P[n - 1, x], x])/(2 + (2*n + 3)*x + (n^2 + n)*x^2)]];\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] = ((4 + (20*n - 10)*x + (18*n^2 + 10*n - 24)*x^2 + (4*n^3 + 8*n^2 - 4*n - 8)*x^3)*v[i - 1] - (8*x + (12*n + 32)*x^2 + (4*n^2 + 36*n + 40)*x^3 + (8*n^2 + 24*n + 16)*x^4)*deriv(v[i - 1], x))/(2 + (2*n + 3)*x + (n^2 + n)*x^2)); 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-b6aca576a689e9f0","steps":["basis A013611","\u03b8+1"],"edges":["e-6f6524a2909f4dfb","e-3de5ce8dcab4eeb0"],"text":"A274496 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A274496 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}
