{"id":"n-f2595331a852a6f0","revision":1,"first_index":0,"fingerprint":"f6-ec9d723a8ce3d936","name":"theta_plus_one_squared of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","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","4"],["1","8"],["1","12","9"],["1","16","27"],["1","20","54","16"],["1","24","90","64"],["1","28","135","160","25"],["1","32","189","320","125"],["1","36","252","560","375","36"],["1","40","324","896","875","216"],["1","44","405","1344","1750","756","49"],["1","48","495","1920","3150","2016","343"],["1","52","594","2640","5250","4536","1372","64"],["1","56","702","3520","8250","9072","4116","512"]],"degrees":[0,1,1,2,2,3,3,4,4,5,5,6,6,7,7],"row_sums":["1","5","9","22","44","91","179","349","667","1260","2352","4349","7973","14509","26229"],"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":[["2","1","2"],["3","7","7"],["1","9","9"],["0","3","5"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["2","1","2","0"],["3","23","39","36"],["1","17","47","42"],["0","3","15","16"],["0","0","1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-19","-62","-72"],["0","-1","-16","-69","-84"],["0","0","-3","-20","-32"],["0","0","0","-1","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160894,"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-cd5540b5c0539e97"]},{"id":160895,"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-cd5540b5c0539e97"]},{"id":160896,"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-cd5540b5c0539e97"]}],"region_exact":{},"incoming_edges":[{"id":"e-cd5540b5c0539e97","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-f2595331a852a6f0","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":"abed7244cea850d58d3da4bd754878c4c29aa5b05f852be330eba7f31e54708c","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-569a801b809b6abd","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","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-49efffb3783e4c31","kind":"fingerprint_related","node_a_id":"n-ed2256bce5038d1b","node_b_id":"n-f2595331a852a6f0","fingerprint":"f6-ec9d723a8ce3d936","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ec9d723a8ce3d936"},"other_id":"n-ed2256bce5038d1b","other":{"id":"n-ed2256bce5038d1b","name":"theta_plus_one_squared of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","first_index":1,"fingerprint":"f6-ec9d723a8ce3d936","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","8"]],"row_sums_preview":["1","1","5","9","22","44","91","179"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8b3f39d03de798f","steps":["(\u03b8+1)\u00b2"],"edges":["e-34d5cc3a968e30b1"],"text":"A052553 \u2192 (\u03b8+1)\u00b2"},"headline":"A052553 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-bee7dbacc9ccb12c","kind":"fingerprint_related","node_a_id":"n-14012eded861244a","node_b_id":"n-f2595331a852a6f0","fingerprint":"f6-ec9d723a8ce3d936","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ec9d723a8ce3d936"},"other_id":"n-14012eded861244a","other":{"id":"n-14012eded861244a","name":"theta_plus_one_squared of Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","first_index":0,"fingerprint":"f6-ec9d723a8ce3d936","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","8"]],"row_sums_preview":["1","1","5","9","22","44","91","179"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75ccd0453a9b6cec","steps":["(\u03b8+1)\u00b2"],"edges":["e-268a0da4668d8a40"],"text":"A011973 \u2192 (\u03b8+1)\u00b2"},"headline":"A011973 \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,  4;\n  2: 1,  8;\n  3: 1, 12,   9;\n  4: 1, 16,  27;\n  5: 1, 20,  54,  16;\n  6: 1, 24,  90,  64;\n  7: 1, 28, 135, 160,  25;\n  8: 1, 32, 189, 320, 125;\n  9: 1, 36, 252, 560, 375, 36;\n","recurrence_sentence":"The row polynomials satisfy (n^2+3*n+2+(3*n^3+9*n^2+7*n+1)*x+(n^4+5*n^3+9*n^2+7*n+2)*x^2)*P(n,x) = (n^2+3*n+2+(3*n^3+17*n^2+23*n+1)*x+(n^4+15*n^3+47*n^2+39*n+2)*x^2+(2*n^4+16*n^3+42*n^2+36*n)*x^3)*P(n-1,x) - ((n+2)*x+(3*n^2+16*n+19)*x^2+(n^3+20*n^2+69*n+62)*x^3+(4*n^3+32*n^2+84*n+72)*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[((n^2 + 3*n + 2 + (3*n^3 + 17*n^2 + 23*n + 1)*x + (n^4 + 15*n^3 + 47*n^2 + 39*n + 2)*x^2 + (2*n^4 + 16*n^3 + 42*n^2 + 36*n)*x^3)*P[n - 1, x] - ((n + 2)*x + (3*n^2 + 16*n + 19)*x^2 + (n^3 + 20*n^2 + 69*n + 62)*x^3 + (4*n^3 + 32*n^2 + 84*n + 72)*x^4)*D[P[n - 1, x], x])/(n^2 + 3*n + 2 + (3*n^3 + 9*n^2 + 7*n + 1)*x + (n^4 + 5*n^3 + 9*n^2 + 7*n + 2)*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] = ((n^2 + 3*n + 2 + (3*n^3 + 17*n^2 + 23*n + 1)*x + (n^4 + 15*n^3 + 47*n^2 + 39*n + 2)*x^2 + (2*n^4 + 16*n^3 + 42*n^2 + 36*n)*x^3)*v[i - 1] - ((n + 2)*x + (3*n^2 + 16*n + 19)*x^2 + (n^3 + 20*n^2 + 69*n + 62)*x^3 + (4*n^3 + 32*n^2 + 84*n + 72)*x^4)*deriv(v[i - 1], x))/(n^2 + 3*n + 2 + (3*n^3 + 9*n^2 + 7*n + 1)*x + (n^4 + 5*n^3 + 9*n^2 + 7*n + 2)*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-569a801b809b6abd","steps":["(\u03b8+1)\u00b2"],"edges":["e-cd5540b5c0539e97"],"text":"A169803 \u2192 (\u03b8+1)\u00b2"},"headline":"A169803 \u2192 (\u03b8+1)\u00b2","chain":null}
