{"id":"n-4b901f65c39b0b35","revision":0,"first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","name":"reversal of Triangle of coefficients in expansion of (1+5x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["25","10","1"],["125","75","15","1"],["625","500","150","20","1"],["3125","3125","1250","250","25","1"],["15625","18750","9375","2500","375","30","1"],["78125","109375","65625","21875","4375","525","35","1"],["390625","625000","437500","175000","43750","7000","700","40","1"],["1953125","3515625","2812500","1312500","393750","78750","10500","900","45","1"],["9765625","19531250","17578125","9375000","3281250","787500","131250","15000","1125","50","1"],["48828125","107421875","107421875","64453125","25781250","7218750","1443750","206250","20625","1375","55","1"],["244140625","585937500","644531250","429687500","193359375","61875000","14437500","2475000","309375","27500","1650","60","1"],["1220703125","3173828125","3808593750","2792968750","1396484375","502734375","134062500","26812500","4021875","446875","35750","1950","65","1"],["6103515625","17089843750","22216796875","17773437500","9775390625","3910156250","1173046875","268125000","46921875","6256250","625625","45500","2275","70","1"],["30517578125","91552734375","128173828125","111083984375","66650390625","29326171875","9775390625","2513671875","502734375","78203125","9384375","853125","56875","2625","75","1"],["152587890625","488281250000","732421875000","683593750000","444335937500","213281250000","78203125000","22343750000","5027343750","893750000","125125000","13650000","1137500","70000","3000","80","1"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["36","216","1296","7776","46656","279936","1679616","10077696","60466176","362797056","2176782336","13060694016","78364164096","470184984576","2821109907456"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["25","10","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["5","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185436,"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-3920ec1a803b452d"]},{"id":185437,"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-3920ec1a803b452d"]},{"id":185438,"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-3920ec1a803b452d"]}],"region_exact":{},"incoming_edges":[{"id":"e-3920ec1a803b452d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-4b901f65c39b0b35","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"eeaf8081dbf141737505aeb315620aee7f5f00f09e4c3476c6e2ce7d9f455ca0","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-bbfc5a54ff13a785","role":"sequence","name":"Triangle of coefficients in expansion of (1+5x)^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-02ff04bcd75bc02d","kind":"fingerprint_related","node_a_id":"n-b83665b9a9abd39d","node_b_id":"n-4b901f65c39b0b35","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-b83665b9a9abd39d","other":{"id":"n-b83665b9a9abd39d","name":"mobius_x_over_x_plus_one of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","5"],["1","10","25"]],"card_rows":[["1"],["1","5"],["1","10","25"],["1","15","75","125"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["x \u21a6 x/(x+1)"],"edges":["e-949a223514c284da"],"text":"A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-355504a3f75981e6","kind":"fingerprint_related","node_a_id":"n-4b901f65c39b0b35","node_b_id":"n-1125779e0b7ca410","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-1125779e0b7ca410","other":{"id":"n-1125779e0b7ca410","name":"reversal of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["9","90","225"],["27","405","2025","3375"],["81","1620","12150","40500","50625"]],"card_rows":[["9","90","225"],["27","405","2025","3375"],["81","1620","12150","40500","50625"],["243","6075","60750","303750","759375","759375"]],"row_sums_preview":["324","5832","104976","1889568","34012224","612220032","11019960576","198359290368"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["x \u21a6 x+1","reverse"],"edges":["e-5083e5230277d126","e-86f80e580943670b"],"text":"A038329 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A038329 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-3807a1a67427de7a","kind":"fingerprint_related","node_a_id":"n-4b901f65c39b0b35","node_b_id":"n-9cce4140bcdbe78c","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-9cce4140bcdbe78c","other":{"id":"n-9cce4140bcdbe78c","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["400","160","16"],["8000","4800","960","64"],["160000","128000","38400","5120","256"]],"card_rows":[["400","160","16"],["8000","4800","960","64"],["160000","128000","38400","5120","256"],["3200000","3200000","1280000","256000","25600","1024"]],"row_sums_preview":["576","13824","331776","7962624","191102976","4586471424","110075314176","2641807540224"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A007318","basis A007318"],"edges":["e-7b68ca495e98f7ea","e-70a0512cff30529a"],"text":"A038330 \u2192 basis A007318 \u2192 basis A007318"},"headline":"A038330 \u2192 basis A007318 \u2192 basis A007318","chain":null}},{"id":"l-392c7334d4158e1d","kind":"fingerprint_related","node_a_id":"n-4b901f65c39b0b35","node_b_id":"n-c66fb08c4dfad91a","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-c66fb08c4dfad91a","other":{"id":"n-c66fb08c4dfad91a","name":"derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["90","18"],["2025","810","81"],["40500","24300","4860","324"]],"card_rows":[["90","18"],["2025","810","81"],["40500","24300","4860","324"],["759375","607500","182250","24300","1215"]],"row_sums_preview":["108","2916","69984","1574640","34012224","714256704","14693280768","297538935552"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["x \u21a6 x+1","D"],"edges":["e-5083e5230277d126","e-47539391ec5f1610"],"text":"A038329 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A038329 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-3bb73730fd39e063","kind":"fingerprint_related","node_a_id":"n-0aa4a1e9e961a941","node_b_id":"n-4b901f65c39b0b35","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-0aa4a1e9e961a941","other":{"id":"n-0aa4a1e9e961a941","name":"derivative of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["10","50"],["15","150","375"],["20","300","1500","2500"]],"card_rows":[["10","50"],["15","150","375"],["20","300","1500","2500"],["25","500","3750","12500","15625"]],"row_sums_preview":["60","540","4320","32400","233280","1632960","11197440","75582720"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["D"],"edges":["e-8f637c73fe2c30f0"],"text":"A013612 \u2192 D"},"headline":"A013612 \u2192 D","chain":null}},{"id":"l-4a33962242984782","kind":"fingerprint_related","node_a_id":"n-4b901f65c39b0b35","node_b_id":"n-b9fb3905119e37a6","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-b9fb3905119e37a6","other":{"id":"n-b9fb3905119e37a6","name":"reversal of basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","15"],["9","90","225"]],"card_rows":[["1"],["3","15"],["9","90","225"],["27","405","2025","3375"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013610","reverse"],"edges":["e-011a72a323f5e83e","e-0da40532f4b4c25e"],"text":"A147716 \u2192 basis A013610 \u2192 reverse"},"headline":"A147716 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-4f0ad1596c9a461b","kind":"fingerprint_related","node_a_id":"n-4b901f65c39b0b35","node_b_id":"n-39158ca46e0de091","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-39158ca46e0de091","other":{"id":"n-39158ca46e0de091","name":"shift_x_plus_one of basis_map_A013611 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["20","4"],["400","160","16"]],"card_rows":[["1"],["20","4"],["400","160","16"],["8000","4800","960","64"]],"row_sums_preview":["1","24","576","13824","331776","7962624","191102976","4586471424"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-9c08db68889738c4","e-176b145beb4f8297"],"text":"A027467 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A027467 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-59d98ffba6f5d33c","kind":"fingerprint_related","node_a_id":"n-2b94985594253b13","node_b_id":"n-4b901f65c39b0b35","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-2b94985594253b13","other":{"id":"n-2b94985594253b13","name":"reversal of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","5"],["1","10","25"]],"card_rows":[["1"],["1","5"],["1","10","25"],["1","15","75","125"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b91a7a037fb3513","steps":["reverse"],"edges":["e-bec3e7775a5cdcbb"],"text":"A038243 \u2192 reverse"},"headline":"A038243 \u2192 reverse","chain":null}},{"id":"l-5f4357d08777ad97","kind":"fingerprint_related","node_a_id":"n-af03db3dc7849a56","node_b_id":"n-4b901f65c39b0b35","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-af03db3dc7849a56","other":{"id":"n-af03db3dc7849a56","name":"mobius_x_over_x_plus_one of basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","10","25"],["1","15","75","125"],["1","20","150","500","625"]],"card_rows":[["1","10","25"],["1","15","75","125"],["1","20","150","500","625"],["1","25","250","1250","3125","3125"]],"row_sums_preview":["36","216","1296","7776","46656","279936","1679616","10077696"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-b91bf4c52fd06745","e-a816678ba2afa32c"],"text":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-68549c97becdaccc","kind":"fingerprint_related","node_a_id":"n-4b901f65c39b0b35","node_b_id":"n-8f7b77214e515821","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-8f7b77214e515821","other":{"id":"n-8f7b77214e515821","name":"derivative of basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["90","18"]],"card_rows":[["3"],["90","18"],["2025","810","81"],["40500","24300","4860","324"]],"row_sums_preview":["0","3","108","2916","69984","1574640","34012224","714256704"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013610","D"],"edges":["e-011a72a323f5e83e","e-3ebd0fbcb1cb9e8c"],"text":"A147716 \u2192 basis A013610 \u2192 D"},"headline":"A147716 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-8a63099aee6a99b3","kind":"fingerprint_related","node_a_id":"n-7b91a7a037fb3513","node_b_id":"n-4b901f65c39b0b35","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-7b91a7a037fb3513","other":{"id":"n-7b91a7a037fb3513","name":"Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","1"],["25","10","1"]],"card_rows":[["1"],["5","1"],["25","10","1"],["125","75","15","1"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A038243"],"derivation":{"root":"n-7b91a7a037fb3513","steps":[],"edges":[],"text":"A038243"},"headline":"A038243: Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","chain":null}},{"id":"l-a5399651c9813ee0","kind":"fingerprint_related","node_a_id":"n-4b901f65c39b0b35","node_b_id":"n-b758fb00deafc303","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-b758fb00deafc303","other":{"id":"n-b758fb00deafc303","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","5"],["2","20","50"]],"card_rows":[["1"],["1","5"],["2","20","50"],["6","90","450","750"]],"row_sums_preview":["1","6","72","1296","31104","933120","33592320","1410877440"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-3656191e13afb7f3","e-6ec3bc6d71fb3107"],"text":"A136572 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A136572 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-b30e8081d4e1ca5f","kind":"fingerprint_related","node_a_id":"n-6323575beb222864","node_b_id":"n-4b901f65c39b0b35","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-6323575beb222864","other":{"id":"n-6323575beb222864","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","2"],["100","40","4"]],"card_rows":[["1"],["10","2"],["100","40","4"],["1000","600","120","8"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013609"],"edges":["e-88ca34e9877f4f47"],"text":"A038291 \u2192 basis A013609"},"headline":"A038291 \u2192 basis A013609","chain":null}},{"id":"l-ba3e03e2b993d525","kind":"fingerprint_related","node_a_id":"n-e3cadb5e194cb15c","node_b_id":"n-4b901f65c39b0b35","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-e3cadb5e194cb15c","other":{"id":"n-e3cadb5e194cb15c","name":"shift_x_plus_one of derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["10","2"]],"card_rows":[["1"],["10","2"],["75","30","3"],["500","300","60","4"]],"row_sums_preview":["0","1","12","108","864","6480","46656","326592"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b91a7a037fb3513","steps":["D"],"edges":["e-37cb40ea1f980993"],"text":"A038243 \u2192 D"},"headline":"A038243 \u2192 D","chain":null}},{"id":"l-c82eba486cba38f0","kind":"fingerprint_related","node_a_id":"n-bbfc5a54ff13a785","node_b_id":"n-4b901f65c39b0b35","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-bbfc5a54ff13a785","other":{"id":"n-bbfc5a54ff13a785","name":"Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","10","25"],["1","15","75","125"],["1","20","150","500","625"]],"card_rows":[["1","10","25"],["1","15","75","125"],["1","20","150","500","625"],["1","25","250","1250","3125","3125"]],"row_sums_preview":["36","216","1296","7776","46656","279936","1679616","10077696"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":[],"edges":[],"text":"A013612"},"headline":"Triangle of coefficients in expansion of (1+5x)^n.","chain":null}},{"id":"l-d370de0b2855e890","kind":"fingerprint_related","node_a_id":"n-804e872f93921cf0","node_b_id":"n-4b901f65c39b0b35","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-804e872f93921cf0","other":{"id":"n-804e872f93921cf0","name":"basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["15","3"],["225","90","9"]],"card_rows":[["1"],["15","3"],["225","90","9"],["3375","2025","405","27"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013610"],"edges":["e-011a72a323f5e83e"],"text":"A147716 \u2192 basis A013610"},"headline":"A147716 \u2192 basis A013610","chain":null}},{"id":"l-d534cdcb61114d10","kind":"fingerprint_related","node_a_id":"n-4b901f65c39b0b35","node_b_id":"n-119c10eed5405419","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-119c10eed5405419","other":{"id":"n-119c10eed5405419","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","10"],["4","40","100"]],"card_rows":[["1"],["2","10"],["4","40","100"],["8","120","600","1000"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-d7d2422771947afc","e-6ecc5a1d7e3e25d3"],"text":"A130123 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A130123 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-d6f8c33e6572b772","kind":"fingerprint_related","node_a_id":"n-4b901f65c39b0b35","node_b_id":"n-69e9b0dfdfe0424c","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-69e9b0dfdfe0424c","other":{"id":"n-69e9b0dfdfe0424c","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","10"],["4","40","100"]],"card_rows":[["1"],["2","10"],["4","40","100"],["8","120","600","1000"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013609","reverse"],"edges":["e-88ca34e9877f4f47","e-6f038ce27cd3a7a7"],"text":"A038291 \u2192 basis A013609 \u2192 reverse"},"headline":"A038291 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-e5708d6bb8982dfb","kind":"fingerprint_related","node_a_id":"n-4b901f65c39b0b35","node_b_id":"n-24bd3442a6d1c2f2","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-24bd3442a6d1c2f2","other":{"id":"n-24bd3442a6d1c2f2","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["40","8"]],"card_rows":[["2"],["40","8"],["600","240","24"],["8000","4800","960","64"]],"row_sums_preview":["0","2","48","864","13824","207360","2985984","41803776"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013609","D"],"edges":["e-88ca34e9877f4f47","e-37c5277fa0460438"],"text":"A038291 \u2192 basis A013609 \u2192 D"},"headline":"A038291 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-f1bcac55e4a3792b","kind":"fingerprint_related","node_a_id":"n-8447c9ce68adee7a","node_b_id":"n-4b901f65c39b0b35","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-8447c9ce68adee7a","other":{"id":"n-8447c9ce68adee7a","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["225","90","9"],["3375","2025","405","27"],["50625","40500","12150","1620","81"]],"card_rows":[["225","90","9"],["3375","2025","405","27"],["50625","40500","12150","1620","81"],["759375","759375","303750","60750","6075","243"]],"row_sums_preview":["324","5832","104976","1889568","34012224","612220032","11019960576","198359290368"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["x \u21a6 x+1"],"edges":["e-5083e5230277d126"],"text":"A038329 \u2192 x \u21a6 x+1"},"headline":"A038329 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-f271e795fc5d5bc4","kind":"fingerprint_related","node_a_id":"n-8b0b17ea58a3d398","node_b_id":"n-4b901f65c39b0b35","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-8b0b17ea58a3d398","other":{"id":"n-8b0b17ea58a3d398","name":"mobius_x_over_x_plus_one of basis_map_A013611 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","5"],["1","10","25"]],"card_rows":[["1"],["1","5"],["1","10","25"],["1","15","75","125"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-f93ad378f9476e41","e-394e87d0ad909faf"],"text":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 x \u21a6 x/(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:       25,        10,         1;\n  1:      125,        75,        15,        1;\n  2:      625,       500,       150,       20,        1;\n  3:     3125,      3125,      1250,      250,       25,       1;\n  4:    15625,     18750,      9375,     2500,      375,      30,       1;\n  5:    78125,    109375,     65625,    21875,     4375,     525,      35,      1;\n  6:   390625,    625000,    437500,   175000,    43750,    7000,     700,     40,     1;\n  7:  1953125,   3515625,   2812500,  1312500,   393750,   78750,   10500,    900,    45,    1;\n  8:  9765625,  19531250,  17578125,  9375000,  3281250,  787500,  131250,  15000,  1125,   50,  1;\n  9: 48828125, 107421875, 107421875, 64453125, 25781250, 7218750, 1443750, 206250, 20625, 1375, 55, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) for n >= 1, with P(0,x) = 25 + 10*x + x^2.","mathematica":"P[0, x_] := 25 + 10*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(5 + x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 25 + 10*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (5 + x)*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-bbfc5a54ff13a785","steps":["reverse"],"edges":["e-3920ec1a803b452d"],"text":"A013612 \u2192 reverse"},"headline":"A013612 \u2192 reverse","chain":null}
