{"id":"n-b9702783978e2f79","revision":0,"first_index":0,"fingerprint":"f6-6cf8c5bbd0a3960a","name":"shift_x_plus_one of veronese_section_2_1 of Exponential Riordan array [x*exp(x),x].","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["2"],["6"],["16","4"],["40","20"],["96","72","6"],["224","224","42"],["512","640","192","8"],["1152","1728","720","72"],["2560","4480","2400","400","10"],["5632","11264","7392","1760","110"],["12288","27648","21504","6720","720","12"],["26624","66560","59904","23296","3640","156"],["57344","157696","161280","75264","15680","1176","14"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","2","6","20","60","174","490","1352","3672","9850","26158","68892","180180","468454"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["2"],["6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","-4"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-4","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172746,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-325cf246e383909b"]},{"id":172747,"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-325cf246e383909b"]},{"id":172748,"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-325cf246e383909b"]}],"region_exact":{},"incoming_edges":[{"id":"e-325cf246e383909b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-b9702783978e2f79","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"d06ee8baa4f17c815074f7bb3a03c65b7ee099553459f67117b7ccbd853d5449","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-0ed898288ff872d1","role":"sequence","name":"veronese_section_2_1 of Exponential Riordan array [x*exp(x),x].","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-05e3da13f6839953","kind":"fingerprint_related","node_a_id":"n-c01c3c5291c87b47","node_b_id":"n-b9702783978e2f79","fingerprint":"f6-6cf8c5bbd0a3960a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6cf8c5bbd0a3960a"},"other_id":"n-c01c3c5291c87b47","other":{"id":"n-c01c3c5291c87b47","name":"shift_x_plus_one of veronese_section_2_1 of binomial","first_index":0,"fingerprint":"f6-6cf8c5bbd0a3960a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["2"]],"card_rows":[["1"],["2"],["4","1"],["8","4"]],"row_sums_preview":["0","1","2","5","12","29","70","169"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["Veronese 2 (j=1)","x \u21a6 x+1"],"edges":["e-3a4f64258abf2a3d","e-6ede56e59a5b328a"],"text":"A007318 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1"},"headline":"A007318 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1","chain":null}},{"id":"l-07547d922955bdb5","kind":"fingerprint_related","node_a_id":"n-5ad70a9d47b9fac2","node_b_id":"n-b9702783978e2f79","fingerprint":"f6-6cf8c5bbd0a3960a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6cf8c5bbd0a3960a"},"other_id":"n-5ad70a9d47b9fac2","other":{"id":"n-5ad70a9d47b9fac2","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.","first_index":0,"fingerprint":"f6-6cf8c5bbd0a3960a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","2"],["0","1","4"]],"card_rows":[["1"],["0","2"],["0","1","4"],["0","0","4","8"]],"row_sums_preview":["1","2","5","12","29","70","169","408"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e8e64448d4c6498a","steps":["x \u21a6 x/(x+1)"],"edges":["e-1acb289ff60ec83d"],"text":"A119900 \u2192 x \u21a6 x/(x+1)"},"headline":"A119900 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-11dd660cc0810cc9","kind":"fingerprint_related","node_a_id":"n-b9702783978e2f79","node_b_id":"n-1c6e7b2945273445","fingerprint":"f6-6cf8c5bbd0a3960a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6cf8c5bbd0a3960a"},"other_id":"n-1c6e7b2945273445","other":{"id":"n-1c6e7b2945273445","name":"basis_map_A013611 of Triangle read by rows: T(n,k) is the number of words over alphabet {0,1,2,3} having exactly k occurrences of the string 01, where n>=0 and k>=0.","first_index":0,"fingerprint":"f6-6cf8c5bbd0a3960a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4"],["16","4"]],"card_rows":[["1"],["4"],["16","4"],["64","32"]],"row_sums_preview":["1","4","20","96","464","2240","10816","52224"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e3a9cbbad5cb7ab4","steps":["basis A013611"],"edges":["e-7948a619be1c7999"],"text":"A261711 \u2192 basis A013611"},"headline":"A261711 \u2192 basis A013611","chain":null}},{"id":"l-14abd0321d5f2277","kind":"fingerprint_related","node_a_id":"n-4f141517bd1aec42","node_b_id":"n-b9702783978e2f79","fingerprint":"f6-6cf8c5bbd0a3960a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6cf8c5bbd0a3960a"},"other_id":"n-4f141517bd1aec42","other":{"id":"n-4f141517bd1aec42","name":"derivative of Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).","first_index":0,"fingerprint":"f6-6cf8c5bbd0a3960a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["1"]],"card_rows":[["1"],["3"],["8","2"],["20","10"]],"row_sums_preview":["0","0","1","3","10","30","87","245"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-495335f6a4ca3289","steps":["D"],"edges":["e-07e653930a487967"],"text":"A201701 \u2192 D"},"headline":"A201701 \u2192 D","chain":null}},{"id":"l-2e1c42679854113f","kind":"fingerprint_related","node_a_id":"n-b2bd543a62b72f70","node_b_id":"n-b9702783978e2f79","fingerprint":"f6-6cf8c5bbd0a3960a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6cf8c5bbd0a3960a"},"other_id":"n-b2bd543a62b72f70","other":{"id":"n-b2bd543a62b72f70","name":"shift_x_plus_one of veronese_section_2_1 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-6cf8c5bbd0a3960a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["4"]],"card_rows":[["2"],["4"],["8","2"],["16","8"]],"row_sums_preview":["0","2","4","10","24","58","140","338"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["Veronese 2 (j=1)","basis A007318"],"edges":["e-53a911614b54e844","e-1ef77ab2d533642e"],"text":"A028326 \u2192 Veronese 2 (j=1) \u2192 basis A007318"},"headline":"A028326 \u2192 Veronese 2 (j=1) \u2192 basis A007318","chain":null}},{"id":"l-58a51cd9e8622366","kind":"fingerprint_related","node_a_id":"n-b9702783978e2f79","node_b_id":"n-e7ab1f11d81b9cb5","fingerprint":"f6-6cf8c5bbd0a3960a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6cf8c5bbd0a3960a"},"other_id":"n-e7ab1f11d81b9cb5","other":{"id":"n-e7ab1f11d81b9cb5","name":"shift_x_plus_one of veronese_section_2_1 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-6cf8c5bbd0a3960a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["4"],["20"]],"card_rows":[["4"],["20"],["80","20"],["280","140"]],"row_sums_preview":["0","4","20","100","420","1624","5880","20280"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["Veronese 2 (j=1)","x \u21a6 x+1"],"edges":["e-4dba393820f57c76","e-29f1f7a533a73886"],"text":"A178820 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1"},"headline":"A178820 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1","chain":null}},{"id":"l-75beabf1a5b4e56c","kind":"fingerprint_related","node_a_id":"n-da91753b522836dd","node_b_id":"n-b9702783978e2f79","fingerprint":"f6-6cf8c5bbd0a3960a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6cf8c5bbd0a3960a"},"other_id":"n-da91753b522836dd","other":{"id":"n-da91753b522836dd","name":"shift_x_plus_one of Triangle of odd-numbered terms in rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-6cf8c5bbd0a3960a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["4","1"]],"card_rows":[["1"],["2"],["4","1"],["8","4"]],"row_sums_preview":["1","2","5","12","29","70","169","408"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fd133d9d2402ed18","steps":["x \u21a6 x+1"],"edges":["e-c5808b2b5ec8df5e"],"text":"A034867 \u2192 x \u21a6 x+1"},"headline":"A034867 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-79bc546dea68118c","kind":"fingerprint_related","node_a_id":"n-830cf6b474af0428","node_b_id":"n-b9702783978e2f79","fingerprint":"f6-6cf8c5bbd0a3960a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6cf8c5bbd0a3960a"},"other_id":"n-830cf6b474af0428","other":{"id":"n-830cf6b474af0428","name":"shift_x_plus_one of veronese_section_2_1 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-6cf8c5bbd0a3960a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[[],["-1"],["2"]],"card_rows":[["-1"],["2"],["-4","-1"],["8","4"]],"row_sums_preview":["0","-1","2","-5","12","-29","70","-169"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["Veronese 2 (j=1)","basis A007318"],"edges":["e-a70814c31408551f","e-6264503eda6bd957"],"text":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 (j=1) \u2192 basis A007318"},"headline":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 (j=1) \u2192 basis A007318","chain":null}},{"id":"l-bdcc4388c2ae5803","kind":"fingerprint_related","node_a_id":"n-b9702783978e2f79","node_b_id":"n-834a0f9951812ecb","fingerprint":"f6-6cf8c5bbd0a3960a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6cf8c5bbd0a3960a"},"other_id":"n-834a0f9951812ecb","other":{"id":"n-834a0f9951812ecb","name":"shift_x_plus_one of veronese_section_2_1 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-6cf8c5bbd0a3960a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["8"]],"card_rows":[["2"],["8"],["32","8"],["128","64"]],"row_sums_preview":["0","2","8","40","192","928","4480","21632"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["Veronese 2 (j=1)","basis A007318"],"edges":["e-0d347b78871c0570","e-0fb7840525e1618e"],"text":"A038208 \u2192 Veronese 2 (j=1) \u2192 basis A007318"},"headline":"A038208 \u2192 Veronese 2 (j=1) \u2192 basis A007318","chain":null}},{"id":"l-c4215f7f08e03c3b","kind":"fingerprint_related","node_a_id":"n-b9702783978e2f79","node_b_id":"n-46bae8d0b3f1be4f","fingerprint":"f6-6cf8c5bbd0a3960a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6cf8c5bbd0a3960a"},"other_id":"n-46bae8d0b3f1be4f","other":{"id":"n-46bae8d0b3f1be4f","name":"shift_x_plus_one of veronese_section_2_1 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-6cf8c5bbd0a3960a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["5"],["30"]],"card_rows":[["5"],["30"],["140","35"],["560","280"]],"row_sums_preview":["0","5","30","175","840","3654","14700","55770"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["Veronese 2 (j=1)","x \u21a6 x+1"],"edges":["e-1471988cf5f0d7aa","e-01b16dd7d0871a99"],"text":"A178821 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1"},"headline":"A178821 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1","chain":null}},{"id":"l-c8ae2e064163e31f","kind":"fingerprint_related","node_a_id":"n-b9702783978e2f79","node_b_id":"n-b92815c7f855e2d1","fingerprint":"f6-6cf8c5bbd0a3960a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6cf8c5bbd0a3960a"},"other_id":"n-b92815c7f855e2d1","other":{"id":"n-b92815c7f855e2d1","name":"shift_x_plus_one of veronese_section_2_1 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-6cf8c5bbd0a3960a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["6"],["42"]],"card_rows":[["6"],["42"],["224","56"],["1008","504"]],"row_sums_preview":["0","6","42","280","1512","7308","32340","133848"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["Veronese 2 (j=1)","x \u21a6 x+1"],"edges":["e-d44b85edec7c5853","e-7ef5306d01c894cd"],"text":"A178822 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1"},"headline":"A178822 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+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:    0;\n  1:    0;\n  2:    2;\n  3:    6;\n  4:   16,    4;\n  5:   40,   20;\n  6:   96,   72,   6;\n  7:  224,  224,  42;\n  8:  512,  640, 192,  8;\n  9: 1152, 1728, 720, 72;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - 4*x*P(n-3,x) - (4-2*x)*P(n-2,x) + 4*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2, P(3,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[3, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - 4*x*P[n - 3, x] - (4 - 2*x)*P[n - 2, x] + 4*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; v[4] = 6; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - 4*x*v[i - 3] - (4 - 2*x)*v[i - 2] + 4*v[i - 1]); 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-7b5887f058255aa1","steps":["Veronese 2 (j=1)","x \u21a6 x+1"],"edges":["e-888e3b1c2eb8e007","e-325cf246e383909b"],"text":"A003506 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1"},"headline":"A003506 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1","chain":null}