{"id":"n-602e5f94214b4713","revision":0,"first_index":1,"fingerprint":"f6-88dd3d68aa6e666a","name":"theta_plus_one of shift_x_plus_one of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["2","2"],["3","4"],["5","10","3"],["8","20","9"],["13","40","27","4"],["21","76","66","16"],["34","142","153","56","5"],["55","260","333","160","25"],["89","470","699","420","100","6"],["144","840","1422","1024","325","36"],["233","1488","2826","2376","950","162","7"],["377","2616","5508","5296","2555","588","49"],["610","4570","10566","11440","6475","1890","245","8"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","4","7","18","37","84","179","390","833","1784","3791","8042","16989","35804"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-10","-2","0"],["10","-3","1"],["0","5","-2"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["-50","-60","-16","0"],["30","-4","-27","-4"],["0","20","12","-2"],["0","0","3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-25","-55","-38","-8","0"],["0","-25","-50","-29","-4"],["0","0","-5","-9","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171559,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-e983e6f643e02cf8"]},{"id":171560,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-e983e6f643e02cf8"]},{"id":171561,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-e983e6f643e02cf8"]}],"region_exact":{},"incoming_edges":[{"id":"e-e983e6f643e02cf8","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-602e5f94214b4713","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":"00e28b0db1428e2023064b7a4801008ad9af904ae7cad827db3ae4c780537da1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-622af52a4165c278","role":"sequence","name":"shift_x_plus_one of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-4d8bae7f961b758b","kind":"fingerprint_related","node_a_id":"n-5ae50734438934fd","node_b_id":"n-602e5f94214b4713","fingerprint":"f6-88dd3d68aa6e666a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-88dd3d68aa6e666a"},"other_id":"n-5ae50734438934fd","other":{"id":"n-5ae50734438934fd","name":"theta_plus_one of Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).","first_index":0,"fingerprint":"f6-88dd3d68aa6e666a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","2"]],"card_rows":[["1"],["1"],["2","2"],["3","4"]],"row_sums_preview":["1","1","4","7","18","37","84","179"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7d332fcb7bf9a0f7","steps":["\u03b8+1"],"edges":["e-6733a977e0405d2b"],"text":"A128100 \u2192 \u03b8+1"},"headline":"A128100 \u2192 \u03b8+1","chain":null}},{"id":"l-c4a5ed5ba92eb9e1","kind":"fingerprint_related","node_a_id":"n-602e5f94214b4713","node_b_id":"n-29cbe2407e2e77bc","fingerprint":"f6-88dd3d68aa6e666a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-88dd3d68aa6e666a"},"other_id":"n-29cbe2407e2e77bc","other":{"id":"n-29cbe2407e2e77bc","name":"theta_plus_one of shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","first_index":0,"fingerprint":"f6-88dd3d68aa6e666a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["3","4"]],"card_rows":[["1"],["2","2"],["3","4"],["5","10","3"]],"row_sums_preview":["1","4","7","18","37","84","179","390"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-569a801b809b6abd","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-124c90dc6fe1d800","e-7b65c31909a3d1ae"],"text":"A169803 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A169803 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  1;\n   2:  1;\n   3:  2,   2;\n   4:  3,   4;\n   5:  5,  10,   3;\n   6:  8,  20,   9;\n   7: 13,  40,  27,   4;\n   8: 21,  76,  66,  16;\n   9: 34, 142, 153,  56,  5;\n  10: 55, 260, 333, 160, 25;\n","recurrence_sentence":"The row polynomials satisfy (10*n-10+(5*n^2-3*n-2)*x+(n^3-2*n^2+n)*x^2)*P(n,x) = (30*n-50+(20*n^2-4*n-60)*x+(3*n^3+12*n^2-27*n-16)*x^2+(2*n^3-2*n^2-4*n)*x^3)*P(n-1,x) - (25+(25*n+55)*x+(5*n^2+50*n+38)*x^2+(9*n^2+29*n+8)*x^3+(4*n^2+4*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((30*n - 50 + (20*n^2 - 4*n - 60)*x + (3*n^3 + 12*n^2 - 27*n - 16)*x^2 + (2*n^3 - 2*n^2 - 4*n)*x^3)*P[n - 1, x] - (25 + (25*n + 55)*x + (5*n^2 + 50*n + 38)*x^2 + (9*n^2 + 29*n + 8)*x^3 + (4*n^2 + 4*n)*x^4)*D[P[n - 1, x], x])/(10*n - 10 + (5*n^2 - 3*n - 2)*x + (n^3 - 2*n^2 + n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((30*n - 50 + (20*n^2 - 4*n - 60)*x + (3*n^3 + 12*n^2 - 27*n - 16)*x^2 + (2*n^3 - 2*n^2 - 4*n)*x^3)*v[i - 1] - (25 + (25*n + 55)*x + (5*n^2 + 50*n + 38)*x^2 + (9*n^2 + 29*n + 8)*x^3 + (4*n^2 + 4*n)*x^4)*deriv(v[i - 1], x))/(10*n - 10 + (5*n^2 - 3*n - 2)*x + (n^3 - 2*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c8b3f39d03de798f","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-13fa50edf65b735e","e-e983e6f643e02cf8"],"text":"A052553 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A052553 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}
